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Electrical Measurements 



A LABORATORY MANUAL 



BY 

HENRY S: CARHART, M.A., LL.D. 

PROFESSOR OF PHYSICS 
AND 

GEORGE Wr PATTERSON, JR., M.A., B.S. 

ASSISTANT PROFESSOR OF PHYSICS 
UNIVERSITY OF MICHIGAN 



Boston 

ALLYN AND BACON 
1895 




Copyright, i8qs, 

BY 

Henry S. Carhart 

AND 

George W. Patterson, Jr. 



<^ 



PRESS or 






PREFACE. 



Progress in the methods of Electrical Measurement 
is quite as marked as in the applications of electricity. 
The perfecting of measuring instruments keeps pace 
with the demands imposed by scientific accuracy. 
Laboratory practice should not be permitted to lag 
behind discovery and commercial applications ; obso- 
lete methods may with propriety be relegated to 
historical collections, along with antiquated apparatus, 
so that students in electricity may learn only the latest 
modes of procedure. 

The authors of this book have proceeded on this 
plan in collecting and devising methods to form a 
graded series of experiments for the use of several 
classes in electrical measurements. How well they 
have succeeded others must decide. Quantitative 
experiments only have been introduced, and they have 
been selected with the object of illustrating the general 
methods of measurement rather than the applications to 
specific departments of technical work, such as submarine 
cable testing, telegraphy and telephony, or d3aiamo 
electric machinery. It is thought to be better that these 
subjects should be treated in special handbooks. 

It is assumed that electrodynamometers and direct 
reading ammeters and voltmeters of good quality are 
now a part of every laboratory equipment, and methods 
are given for their ready calibration. Much less space 



IV PREFACE. 

has been devoted to the tangent galvanometer than 
has been customary in the past; but it has been retained 
because it is a good appliance for practice, though very 
inferior as an instrument of precision in comparison 
with later instruments for measuring current. Zero 
methods have been resorted to wherever it has appeared 
practicable to do so. The student is advised to use 
them as far as possible. 

The experience of a number of years leads to the 
conclusion that the Standard Cell may be made of very 
great service in electrical measurements. Its construc- 
tion has therefore been described with a good deal of 
detail, and a considerable number of experiments involv- 
ing its use have been introduced. Since the Clark cell 
is now the legal standard of electromotive force, both in 
Great Britain and the United States, its use should be 
encouraged for this reason, aside from its convenience. 

The several chapters have been introduced in what 
appears to the authors the order of the difficulties 
involved in them. Further, in each chapter the simpler 
experiments have been described first, and the more 
difficult ones later on. It is assumed that the student 
has completed a first course in the principles of Physics, 
and that he has some knowledge of analytic geometry 
and the calculus. It will be found of advantage if he 
has also had a course in the physical laboratory, com- 
prising measurements of length, mass, periods of oscilla- 
tion, moments of inertia, and the like. 

It will be noticed also that we have not contented 
ourselves with the description of methods, but have 
added an explanation or a demonstration of the principle 
involved, and have given numerous references to orig- 
inal sources of information. 



PREFACK V 

The subject of induction coefficients has been treated 
with more detail than usual on account of the increas- 
ing interest in it in connection with alternating currents 
and their practical applications. Dr. Karl E. Guthe, 
Instructor in Physics, has kindly determined by experi- 
ment the practical details of seyeral of the methods 
described. 

It is hoped that the examples, which for the most part 
haye been taken from work done under the super yision of 
the authors, will proye a useful feature of the manual. 

Thanks are due to Nalder Brothers & Co., Queen & 
Co., and the Weston Electrical Instrument Co., for 
kmdly furnishing a number of the illustrations of ap- 
paratus made by them. 

University of Michigan, 1895. 



CONTENTS 



Chapter Page 

I. Definition of Units and their Dimensional Formulas 1 

II. Resistance 20 

in. Measdrejient or Current ...... 118 

IV. Measurement of Electromotive Force . . . 176 

V. Quantity and Capacity 207 

VI. Self-Induction and Mutual Induction . . . 235 

VII. Magnetism 275 

Appendix A ........ . 321 

Appendix B 328 

Index 337 



ELECTRICAL MEASUREMENTS, 



CHAPTER I. 

DEFINITIONS OF UNITS AND THEIR DIMENSIONAL 
FORMULAS. 

1. Fundamental and Derived Units. — One kind of 
quantity may always be expressed in terms of two or 
three other kinds. For example : Velocity, involving 
two other kinds ; force, invohdng three other quantities. 

A systematic scheme of units involves as many differ- 
ent ones as there are kinds of quantity to be measured ; 
and it connects them together, at least in all dynamic 
science, in such a manner that they are defined in terms 
of three original or underived units. The three which 
are generally employed for this purpose are the units of 
length, time, and mass. These are called fundamental 
units, in distinction from all others, which in turn are 
called derived units. This particular selection is a 
matter of convenience rather than of necessity, and rests 
upon several considerations which properly determine 
the selection of these fundamental quantities. 

2. Dimensional Formulas. — In all scientific inves- 
tigations of a quantitative character it is of great impor- 
tance to know the relations of the derived units to 
the fundamentals ; so that whatever arbitrary units are 
employed as the fundamentals, it may be possible to pass 



2 ELECTRICAL MEASUREMENTS. 

directly and Avith certainty from one system of arbitrary 
fundamentals to another. Tliis is most conveniently 
done by expressing the dimensions of all units. Dimen- 
sional formulas show the powers of the fundamentals that 
enter into the derived units. When a given unit varies 
as the n'^ power of a fundamental, it is said to be of n 
dimensions with respect to that fundamental. Thus the 
unit of area is of two dimensions as regards a length, 
while the unit of volume is of three dimensions with 
respect to the linear unit employed. In other words, the 
unit of area varies as the square of the unit of length, 
and the unit of volume as the third power of the same. 
''Every expression for a quantity consists of two fac- 
tors or components. One of these is the name of a cer- 
tain known quantity of the same kind as the quantity to 
be expressed, which is taken as a standard of reference." ^ 
The other is merely numerical, and expresses the num- 
ber of times the standard must be applied to make up 
the quantity measured. Thus (ten) (feet), (five) 
(grammes), (fifty) (seconds). The dimensions of a 
length are simply L ; of time, T ; and of mass, M.^ The 
numerical part of an expression does not enter into the 
dimensional equation. It is exactly these numerical rela- 
tions that we wish to determine by means of the dimen- 
sional formulas, when we have occasion to pass from one 
system of fundamentals to another. Thus, if we have 
given the numerical constants of an equation expressing 
the relation between any physical quantities, with the 
foot, the pound, and the second as the three arbitrary 
fundamental units, to find the numerical constants of 
the same relation with the centimetre, the gramme, and 

^ Maxwell's Electncity and Magnetism, p. 1. 

2 They arc sometimes written with a square bracket aud sometimes without. 



DEFINITIONS OF UNITS. 8 

the second as the arbitrary fundamentals, we need to 
know only the ratios between the three pairs of funda- 
mentals and the relation of the derived units to the fun- 
damentals, or the dimensional formulas of those derived 
units which express the given physical relationship. 

Further, it is important to observe that the numerical 
parts of two expressions for the same quantity in differ- 
ent units are inversely as the magnitudes of the units 
employed. Thus, if L [X] represents a given linear 
quantity in feet and I [Z] the same quantity in metres, 
in which the parts enclosed iii brackets are the units of 
length, the foot and the metre respectively, then 
liq=L [i], or 

Since [Z] = 3.280856 [X] (one metre = 3.280856 feet) it 
follows that 

L = 3.280856 I 

3. Examples of the Use of Dimensional Formulas. 

— First. A pendulum with a mass of 1 kg. has an equiva- 
lent length of 1 m. Its moment of inertia in cm.'^ — gwi. is 
1000 X 1002 ^ 10^ 



What is it in mm.^ — mg. 



1 mm. = Yo ^"i^^- 

1 mg. = loVo ^^• 
Hence 1 cm.^ = 1 mm.^x 10^ 

and 1 gm. = 1 mg. x 10^. 

Hence 1 cm.^ —gm. = 1 mm.^ — mg. x 10^. 

Since the numerical part of an expression for a given 
quantity is inversely as the magnitude of the unit of 
measurement, it follows that 

10^ em.- — gm. = W x 10'' mm.' — vig. = 10^^ 



4 ELECTRICAL MEASUREMENTS, 

Second. The period of vibration of a pendulum de- 
pends on its length and on gravity. Let us assume that 
it varies as the m*^ power of its length and as the 7i^^ 
power of g. 

Then since gravity is an acceleration, which is the 
rate of change of velocity, and velocity is a length 
divided by a time, it follows that acceleration is a length 
divided by the second power of a time. We may there- 
fore write the dimensional equation for the period of 
vibration of a pendulum in accordance with the assumed 
relationship, thus: 

But the dimensions of the terms in both members of the 

equation must be identical. On one side we have T, 

and on the other T—^''. 

Hence l = — 2n 

1 
or n=^ —-. 

A 

Also = '??^ + 71 = 772 — - , and m = ^. 

Hence the time of vibration of a pendulum varies 
directl}' as the square root of its length, and inversely as 
the square root of gravity, 



or T= const 



VJ 



4. The Unit of Length Nearly all the quantities 

with which physical science deals are measured in units 
which in practice are referred to the three fundamental 
units of length, mass, and time, irrespective of the par- 
ticular system to which these three units belong. But 



BEFIXITIOKS OF UNITS. 5 

it is eminently desirable to so choose these standards 
as fundamentals that we shall have a systematic arrange- 
ment, avoiding numerous and fractional ratios. The 
variety of weights and measures employed commercially 
in the United States and England illustrates an unsys- 
tematic arrangement. The metric system, on the other 
hand, is an example of a logical and simple system- 
atic arrangement and relationship of the various units 
employed. Hence the metric system is now almost 
exclusively used in science. 

Theoretically the metre was intended to be the ten- 
millionth part of the earth-quadrant passing through 
Paris from the equator to the north pole. Practically 
the metre is the distance between the ends of a bar of 
platinum when at 0° C, preserved in the national archives 
at Paris, and known as the Metre des Archives. This 
bar was made by Borda. It was constructed in accord- 
ance with a decree of the French Republic, passed in 
1795, on the recommendation of a committee of the 
Academy of Sciences, consisting of Laplace, Delambre, 
Borda, and others. The arc of a meridian between 
Dunkirk and Barcelona was measured by Delambre and 
Mechain, and the length of the metre was derived from 
this measurement. An earth-quadrant is now known 
to be about 

10,002,015 metres. 

The relation between the foot and the metre is 

1 metre = 3.280856 ft. 

By Act of Congress of the United States, in 1866, the 
metre was defined to be 39.37 inches. The imit of 
length employed in magnetic and electrical measure- 
ments is the y^ part of a metre, called a centimetre. 



6 ELECTRICAL MEASUREMENTS. 

The choice of the centimetre was made by the British 
Association Committee on Electrical Standards and 
Measurements. 

5. The Unit of Mass. — It is important to distin- 
guish between mass and weight. Mass is the quantity 
of matter contained in a body. It is entirely independ- 
ent of gra\dty, though gravity is usually employed to 
compare masses. Weight, on the other hand, means the 
downward force of gravity on a bod}', and is measured 
by gravity. Weight depends upon the situation of a 
body on the earth, and is the product of mass and grav- 
ity. Hence the weight of a given mass of matter varies 
with the variation of gravity from place to place. 

Theoretically the unit of mass in the C.G.S. system 
is the gramme, or the mass of a centimetre cube of 
distilled water at the temperature of maximum density, 
or 4° C. Practically it is the xoVo P^^^ of a standard 
mass of platinum preserved in the archives at Paris, and 
called the Kilogramme des Arcliives. This, also, was 
made by Borda in accordance with the decree of 1795. 
The theoretical and practical definitions prove not to be 
absolutely identical. 

From Kupffer's observations Miller deduces the abso- 
lute density of water as 1.000013.^ Hence the practical 
kilogramme is defined not as the mass of a cubic deci- 
metre of distilled water at 4° C, but as the kilogramme 
of Borda, though the two are very approximately equal. 

The gramme was recommended as the unit of mass by 
the British Association Committee because of its con- 
venience, since it is nearly the mass of unit volume of 

1 According to the obsei-vations of Trallis, reduced by Broch, it is 0.99988. 
— Everett, C.G.S. System of Units, p. 34. 



DEFI^UTIO^^S OF UNITS. 1 

water at maximum density ; and as water is usually 
taken as the standard in determining specific gravity, 
it follows that densities and specific gravities become 
numerically equal. 

6. The Unit of Time. — The unit of time univer- 
sally employed in scientific investigations is the second 
of mean solar time. An apparent solar day is the in- 
terval between two successive transits of the sun's centre 
across the meridian of any place. But since the appar- 
ent solar day varies in length from day to day by reason 
of the unequal velocity of the earth in its orbit, the 
mean or average length of all the apparent solar days 
throughout the year is taken and divided into 86,400 
equal parts, each of which is a second of mean solar 
time. 

7. Dimensions of Mechanical Units. — Area. Since 
area is a length multiplied by a length, its dimensional 
formula is L^. 

Volume. Since volume is a length or space of three 
dimensions, its dimensional formula is L^. 

Velocity. Velocity is a length divided by a time, or 

generally — 

Hence its dimensions are y—LT-^. 

Acceleration. Acceleration is the time-rate of change 

of velocity, or — - . Its dimensional formula is therefore 

at 

LT-'^T^LT-\ 

Force. The magnitude of a force is the product of 



8 ELECTRICAL MEASUREMENTS. 

mass by acceleration. Hence the dimensional equation 
for force is 

F^:: Mx LT- - = LMT- \ 

If, therefore, the unit of time should be changed from 
the second to the minute, the unit of force would be 
reduced to 1/60^ or 1/3600. 

Momentum. Momentum is the product of mass and 
velocity. Its dimensional formula is 

MxLT-^^MLT-\ 

Force, according to Gauss, is measured by the time- 
rate of change of momentum. Its dimensions should 
then be 

MLT-^^ T=MLT-\ 

the same as before. 

The unit of force in the C.G.S. system is that force 
which acting on a gramme mass for one second imparts 
to it a velocity of one cm. per second. This is called 
the dyne. A force of one dyne produces unit accelera- 
tion of unit mass. 

Work. Work is said to be done by a force when it 
produces mass motion in the direction in which the force 
acts. It is numerically equal to the product of the force 
and the component of the displacement produced while 
the force acts, and in the direction in which it acts. The 
dimensions of work are, therefore, a force multiplied by 
a length or 

MLT-'xL:=MUT-\ 

The unit of work in the C.G.S. system is the work 
done by a dyne through one cm. This is called the erg. 
In practical electricity a unit of work, called t\\Q joule., 
and equal to 10' ergs, is frequently used. 



BEFINITIOXS OF UNITS. 9 

Activity. Activity or power is the time-rate of doing 
work. The horse-power in the gravitational system of 
units is a rate of working equal to 33,000 foot-pounds 
per minute, or 550 foot-pounds per second. 

Unit acti^dty in the C.G.S. system is work at the rate 
of one erg per second. The watt., a practical unit of 
activity in electricity, is equal to 10" ergs per second. 
One horse-power is equivalent to 746 watts. 

Since activity is the work done in unit time, its dimen- 
sional formula is 

MUT-'-^T=MrT-K 

Energy is measured by the work done. Its dimensional 
formula is therefore the same as that of work. 

8. Magnetic and Electrical Units. — - Strength of 
Pole. The two ends of a long slender magnet possess 
opposite properties. These ends are called poles., and 
the magnet is said to possess polarity. Poles of the 
same name, sign, or properties repel each other, while 
those possessing opposite properties attract. The strength 
of a pole is accordingly defined as proportional to the 
force it is capable of exerting on another pole. 

If m and m^ represent the strengths of two poles, and 
d is the distance between them, then since magnetic 
attraction and repulsion vary as the inverse square of 
the distance, the force may be expressed as proportional 
to mm' I d?. In the C.G.S. system the constant in the 
expression for/ becomes unity. Unit pole, therefore, has 
unit strength when it repels an equal and'Vmilar pole 
at a distance of one cm. with a force of one dyne. It 
produces unit magnetic field at a distance of one cm. 
from it. 



10 ELECTBICAL MEASUREMENTS. 

We may then write generally 

fd^ = mm' = const, x mm. 

But since constants do not enter into dimensional 
equations, 

m" =^fd^ 

or m =/"2 c?, 

and m = (LMT- 0^ x X = M^I^T- \ 

9. Magnetic Field. — Any region within which a 
magnetic pole is acted upon by magnetic force is called 
a magnetic field. It is a region pervaded by lines of mag- 
netic force, or one in which the ether is in a state of 
strain. 

A magnetic field is completely specified by expressing 
the value and direction of the magnetic force at every 
point. The direction of the force is the line along which 
a positive or north-seeking magnetic pole tends to move, 
and the force is the force sustained by unit pole. If this 
force is called BS^ then the force acting upon any pole 
of strength m is d€m, or 

Hence &8 = -^ > 

m 

The dimensions of 86 are therefore 

MLT- 2 ~ M^L^T- ' = M^L ~ '^T - \ 

Unit magnetic field is one in which a unit magnetic 

pole is acted on by a force of one dyne. 

10. Magnetic Moment. — The product of the 
strength of pole and the length of the magnet is called 
its magnetic mome^it. When a thin magnet of length I 
is placed in a field of strength fiS^ so that it is at right 



DEFINITIO^'S OF UNITS. 11 

angles to the direction of the field, the moment of the 
couple acting on it, tending to turn it so that its mag- 
netic axis shall correspond with the field, is &8ml. When 
the field is imity, this couple becomes ml. Its dimen- 
sional formula is 

11. Intensity of Magnetization. — Intensity of mag- 
netization is the quotient of the magnetic moment of a 
magnet by its volume, or its magnetic moment per cubic 
centimetre. Hence the dimensions of magnetization are 



12. Two Systems of Electrical Units. — A system 
of units for the measurement of any physical quantity 
must be founded upon some phenomenon exhibited by 
the physical agent involved. The two systems of elec- 
trical units in use are founded respectively upon the 
repulsion exhibited by like charges of electricity and 
the magnetic field produced by an electric current. The 
one is therefore called the electrostatic and the other the 
electromagnetic system of units. There is no obvious 
relation between the two, but the dimensional formulas 
of the several units show that the ratio of like units in 
the two systems is either a velocity, the square of a 
velocity, or the reciprocal of the one or the otlier. Many 
series of investigations have been undertaken with a view 
to determine the value of this velocity v. According to 
Maxwell's electromagnetic theory of light, it is numeri- 
cally equal to the velocity of light. At least six different 
methods have been employed with reasonably concurrent 



12 



ELECTBICAL 3IEASURE2IENTS. 



results. The appended table gives a few of the most 
recent values of the ratio v and of the velocity of light : 





Ratio of Units. 


Date. 


Velocity of Light. 


Date. 


Experimenter. 


V in cms. 
per sec. 


Experimenter. 


Vel. of light 
in cms. per sec. 


1883. . 

1888. . 

1889. . 
1889. . 

1889. . 

1890. . 


J. J. Thomson, 
Himstedt . . . 
Rowland . . . 

Rosa 

W. Thomson, 
J. J. Thomson 
and Searle. . 


2.963 XlOio 
3.009 X 10 10 
2.9815 X 1010 
3.0004 X 10 10 
3.004 XlOio 

2.9955 X 1010 


1879. . 
1882 . . 
1882. . 


Michelson . . 
Michelson . . 
jSTewcomb . . 


2.9991 X 1010 
2.9985 X 1010 
2.9981 X 1010 



We shall consider generally only the electromagnetic 
system, founded upon the discovery of Oersted in 1820, 
that a magnetic needle is deflected by an electric current ; 
or, in other words, that a current of electricity produces 
a magnetic field. 



13 . Strength of Current. — A current flowing through 
a loop of wire is equivalent to a magnetic shell, which 
may be considered as composed of a great many short 
filamentary magnets placed side b}^ side, Avith all the 
north-seeking poles forming one surface of the shell, and 
all the south-seeking poles the other surface. The mag- 
netic field at any point produced by a current in an ele- 
ment of the conductor is proportional to the strength 
of the current, to the length of the element, and to 
the inverse square of the distance of the point from 
the element. If we conceive a conductor 1 cm. in 
length, bent into an arc of 1 cm. radius, the current 
through it will have unit strength when it produces 
unit magnetic field at the centre of the arc ; that is, a 
unit pole placed at the centre will be acted on by a force 



DEFINITIONS OF UNITS. 13 

of one dyne at right angles to the plane of the circle. If 
the conductor forms a complete circle of one cm. radius, 
the strength of field at the centre due to unit current 
Tvill be Stt. 

The dimensions of unit current may be derived from 
the consideration that the magnetic field produced by a 
current at the centre of a circular conductor equals the 
strength of the current multiplied by the length of the 
conductor and divided by the square of the radius. Let 
I equal the intensity, or strength, of current. Then 

intensity of field = -^ = BS^ 

or, 1= &8L. 

Hence, 1= M^L -^ T-'xL = M^L^T -^ 

14. Quantity. — The unit of quantity is the quantity 
conveyed by unit current in one second. Its dimen- 
sional formula may, therefore, be found as follows : 

Quantity = current x time 

= M^L^T-'xT 

The unit of quantity is, therefore, independent of the 
unit of time, and depends onl}^ on the units of mass and 
length. 

15. Electromotive Force. — The word force is used 
in this connection in a somewhat figurative way, and not 
in a mechanical sense. 

Force is that which produces or tends to produce 
motion or change of motion of matter. But electro- 
motive force (E.M.F.) produces, or tends to produce, a 



14 ELECTRICAL MEASUREMENTS. 

flow of electricity. It is analogous to hydrostatic press- 
ure, and is often called electric pressure. It must not 
be confused with electric force — a force electrical in 
origin, and producing motion of matter. 

The numerical value of the E.M.F. between two 
points of a circuit, when there is no source of E.M.F. 
in this part of the circuit, equals the difference of poten- 
tial between the same points. Difference of potential 
between two points, A and J5, is defined as the work 
required to be done in carrjdng a unit quantity of elec- 
tricity from the one point to the other. Hence the work 
required to convey a quantity Q from A to B is 

in which Vi and V2 are the potentials of tlie points A and 
B respectively. The electric potential at a point is the 
work required to carry unit electricity from the boundary 
of the field to that point. But since potential difference 
is numerically equal to E.M.F., we have 
E.M.F. r= TF-^ C>. 
Hence the dimensional formula of E.M.F. is 

Unit difference of potential exists between two points 
when one erg of work is expended in conveying unit 
quantity from the one point to the other. 

16. Resistance. — Every conductor of electricity 
offers greater or less obstruction to its passage. The 
researches of Dewar and Fleming^ on the resistance of 
metals at the temperature of boiling oxygen go to show 
that the resistance of all pure metals is zero at — 274° C, 

1 Phil. Mag., Oct., 1892, p. 327 ; Sept., 1893, p. 271. 



i>:efinitions of units. 15 

or the '• absolute zero." The resistance of pure metals 
is, therefore, very nearly proportional to the absolute 
temperature. 

Ohm's law expresses the relation subsisting between 
E.^NI.F., resistance, and current strength. Thus 

where JE expresses the algebraic sum of all the E.M.F.'s 
in the circuit, and B the total resistance. 

XT 

From this i^ — _ , 

or that property of a conductor by virtue of which a part 
of the energy of the current is converted into heat is 
equal to the ratio of the effective E.M.F., producing a 
current, to the current itself. 

A portion ^, ^ of a conductor offers unit resistance 
when the difference of potential between the points A^ 
B is numerically equal to the current produced. 

From the expression for resistance its dimensional 
formula is 

T 

Resistance is, therefore, expressed in terms of a length 
and a time as a velocity. 

17. Capacity. — A conductor possesses unit capacity 
when it is charged by unit quantity to unit difference of 
potential. Since the potential varies directly as the 
charge, we have 



16 ELECTRICAL MEASUREMENTS. 

Ju 

Capacity is, therefore, the reciprocal of an acceleration. 

18. The Practical Electrical Units of the Paris 
Congress of 1881.^ — At the Paris Congress of Elec- 
tricians in 1881, the members of which were otHcially 
delegated by the governments represented, the following 
conclusions were reached: 

1. For electrical measurements the fundamental miits, the 
centimetre, the mass of a gramme, and the second (C.G.S.) shall 
be adopted. 

2. The practical units, the ohm and the volt, shall retain their 
present definitions, 10*^ for the ohm, and 10^ for the volt. 

3. The unit of resistance (ohm) shall be represented by a 
column of mercury of a square millimetre section at the tem- 
perature of zero degrees centigrade. 

4. An international committee shall be charged with the deter- 
mination, by new experiments, for practice of the length of a 
column of mercury of a square millimetre section at the temper- 
ature of zero degrees centigrade, which shall represent the value 
of the ohm. 

5. The current produced by a volt in an ohm shall be called 
the ampere. 

6. The quantity of electricity defined by the condition that an 
ampere gives a coulomb per second shall be called the coulomb. 

7. The capacity defined by the condition that a coulomb in a 
farad gives a volt shall be called the farad. 

19. The Practical Units of the Chicago Congress 
of 1893. — A conference was held at the British Asso- 
ciation meeting in Edinburgh in 1892 in connection with 

1 Congr>.8 International des Electriciens, p. 249. 



DEFINITIONS OF UNITS. 17 

the B. A. Committee on Electrical Standards. In addi- 
tion to members of the committee there were present, 
among others, Professor von Helmholtz, of Germany, 
and M. Guilleaume, of France. At this conference it 
was resolved to adopt the length 106.3 centimetres for 
the mercurial column, and to express the mass of the 
column of constant cross-section instead of the cross- 
sectional area of one square millimetre. These recom- 
mendations the committee on the part of the Board of 
Trade in turn recommended for official adoption by the 
British government. Final official action was, however, 
delayed to await the action of the Chamber of Delegates 
of the International Congress of Electricians, which con- 
vened in Chicago, August 21, 1893.^ 

The following resolutions met the unanimous approval 
of the Chamber : 

Resolved, That the several governments represented by the 
delegates of this International Congress of Electricians be, and 
they are hereby, recommended to formally adopt as legal units 
of electrical measure the following : 

1. As a unit of resistance, the international ohm, which is based 
upon the ohm equal to 10^ units of resistance of the C.G.S. sys- 
tem of electromagnetic units, and is represented by the resist- 
ance offered to an unvarying electric current by a column of 
mercury at the temperature of melting ice, 14.4521 grammes in 
mass, of a constant cross-sectional area and of the length 106.3 
centimetres. 

2. As a unit of current, the international ampere, which is one- 
tenth of the unit of current of the C.G.S. system of electro- 
magnetic units, and which is represented sufficiently well for 
practical use by the unvarying current which, when passed 
through a solution of nitrate of silver in water, in accordance 

1 Proceedings of the Internaiional Electrical Congress, Chicago, 1893 
(Amei\ Inst. Elec. Engineers). 



18 ELECTRICAL MEASUREMENTS. 

with accompanying specification, deposits silver at the rate of 
0.001118 gramme per second. 

3. As a unit of electromotive force, the international volt, 
which is the E.M.F. that, steadily applied to a conductor whose 
resistance is one international ohm, will produce a current of one 
international ampere, and which is represented sufficiently well 
for practical use by ifff of the E.M.F. between the poles or 
electrodes of the voltaic cell known as Clark's Cell, at a temper- 
ature of 15° C, and prepared in the manner described in the 
accompanying specification. 

4. As the unit of quantity, the international coulomb, which is 
the quantity of electricity transferred by a current of one interna- 
tional ampere in one second. 

5. As the unit of capacity, the international farad, which is the 
capacity of a conductor charged to a potential of one international 
volt by one international coulomb of electricity. 

6. As the unit of work, the joule, which is 10^ units of work 
in the C.G.S. system, and which is represented sufficiently well 
for practical use by the energy expended in one second by an 
international ampere in an international ohm. 

7. As the unit of power, the ivatt, which is equal to 10' units of 
power in the C.G.S. system, and which is represented sufficiently 
well for practical use by the work done at the rate of one joule 
per second. 

8. As the unit of induction, the henry, which is the induction 
in the circuit when the E.M.F. induced in this circuit is one inter- 
national volt, while the inducing current varies at the rate of one 
international ampere per second. 

The adoption of these units was approved for publica- 
tion by the Treasury Department of the United States 
government, December 27, 1893. They were made 
legal by Act of Congress, approved by the President, 
July 12, 1894. 

20. Relation between the B.A. Units and the Inter- 
national Units. — The Electrical Standards Committee 
of the British Association for the Advancement of 



DEFINITIONS OF UNITS. 19 

Science has agreed that the following relations exist 
between the B.A. nnit and the international ohm : 

1 B.A. nnit = 0.9866 international ohm. 
1 international ohm = 1.01358 B.A. nnits. 

Since the unit of E.M.F. is defined in terms of the 
ampere and the ohm, and since the ampere is independ- 
ently determined, it follows that the unit of E.M.F. 
varies directly as the unit of resistance. Hence : 

1 B.A. volt = 0.9866 international volt. 
1 international volt = 1.01358 B.A. volts. 

The numeric of any given E.M.F., however, being 
inversely as the value of the unit employed, will have 
reciprocal relations to the above. Thus, if the E.M.F. 
of the Clark normal cell with excess of zinc sulphate 
crystals is 1.434 volts, in B.A. units it is 

1.434 X 1.01358 = 1.453. 

The "legal ohm," which was adopted in 1882 as a 
temporary unit by the international committee, to which 
the subject had been committed by the Congress of 1881, 
was represented by the resistance of a column of mer- 
cury, described as above, but 106 centimetres in length. 
Hence the legal volt and ohm are Yuis ^^ ^^^ corre- 
sponding international units. 



20 ELECTEICAL MEASUREMENTS. 



CHAPTER II. 

RESISTANCE. 

21. The Laws of Resistance. — First. Let AB, 



A B CD 

> 1 1 \ h-> 



Fig. I. 

BC^ CB^ be three resistances, i?i, i?o, jRg, respectively 
(Fig. 1), and let their total resistance in series be R, 
Then is 

Let the potentials of the several points be Fl, Fo, F^, F^. 
Then if I is the current flowing 

V^-Vo = R^I 

Vo-V,= RJ 

V,-V,= RJ 

V,-V, = RL 

These equations are derived from Ohm's law, and are 

true because the current /is the same in each section of 

the conductor. 

By addition of the first three equations, 
V,-V,= QR, + R,+ R.^L 
Combining this with the fourth equation, 

IR = I(R,i-R,+ R,), 
or R = R,^R.2+R,. 



BESISTAyCE. 21 

Hence the resistance of the three conductors placed 
end to end, or m series, is the sum of the resistances of 
the several conductors. If these conductors are parts of 
a uniform wire, it folloAVS that the resistance of a uniform, 
conductor is proportional to its length. This may be 
called the first law of resistance. 

Second. The second law may be derived from a dis- 
cussion of the resistance of parallel circuits. 

Let two conductors of resistance, i^i, R^.^ join two 
points of a circuit A., B. They are then said to be con- 




. Fig. 2. 

nected in parallel or in multiple. Let the potentials of 
the points A and B be Vi and V2 , and let the currents 
through the two branches be Ji and L , the total current 
being I. 

Then by Ohm's law 

Also if R is the combined resistance of the two con- 
ductors in parallel 



1= 



R 



Hence, ^l^:.i^^+i^^ 

R Ri Ri) 

R Ri Ro 



22 ELECTRICAL MEASUBEMENTS. 

The reciprocal of resistance is called 
The conductivity of two conductors in parallel is, there- 
fore, the sum of their separate conductivities. From the 
last equation 



R 



R\ + R2 



This is the expression for the combined resistance of 
the two conductors in parallel. The same reasoning 
may be extended to several conductors in parallel. The 
conductivities of any number of conductors in parallel is 
the sum of their separate conductivities. The resistance 
of three conductors in parallel is 



RiR'2 + RiRz + R\Rz 

If now these resistances are equal to one another, then 
j._ R\ Ri 

These conductors may be considered as elements of a 
single conductor. It follows therefore that the resist- 
ance of a uniform conductor varies inversely as its cross- 
section. 

Thhxl. The specific resistance of a conductor is the 
electrical resistance of a centimetre cube of it wheu 
the current floAvs through from any face to the one oppo- 
site. This is the resistance of a prism of tlie conductor, 
measured from end to end, when the cross-section of the 
prism is a square cm. and the length one cm. Specific 
resistance depends entirely upon the nature of the con- 
ductor. 

Let specific resistance be denoted In' s, nnd let I be 



BESIJSTANCE. "Zd 

the length of a uniform conductor and a its cross-sec- 
tional area. Then its resistance is 

si 
r= - , 
a 

or conversely, s = r-. 

22. The Resistance Temperature Coefficient. — 
The resistance of metallic conductors in general in- 
creases with rise of temperature. If Ito is the resistance 
of a conductor at 0° C, and R^ at f^ then 

R, = Bo(l+at-) 

as a first approximation. In this equation a is the tem- 
perature coefficient, a constant depending upon the 
nature of the conductor. In the case of pure copper 
the extended experiments of Kennelly and Fessenden ^ 
demonstrate a linear relation between the resistance and 
temperature between the limits of 20° C. and 250° C, 
indicating a uniform temperature coefficient of 0.00406 
per degree C. throughout the range. The maximum 
observed value at any point was 0.004097 and the mini- 
mum 0.00399. It is altogether likely that the discre- 
pancies existing among the results obtained b}^ many 
observers should be attributed to the presence of small 
percentages of other metals. 

The temperature coefficient of alloys is in general 
smaller than that of the pure metals comprising them. 
Thus the coefficient of German silver^ composed of 
60 per cent copper, 25.4 per cent zinc, 14.6 per cent 
nickel, is 0.00036, and of platinum-silver, 0.00030. 

1 The Physical Review, Vol. I., p. 260. 

2 Dr. Lindeck, Report of the Electrical Standards Committee of the British 
Association, 1892, p. 9. 



24 



ELECTRICAL MEASUREMENTS. 



The alloy platinoid, consisting of German silver with 
a very small addition of tnngsten, has a coefficient of 
only 0.00022, or abont half that of common German 
silver (0.00044). 

The new alloy, manganin^ composed of 12 per cent of 
manganese, 84 per cent of copper, and about 4 per cent 
of nickel, has a temperature coefficient but slightly in 



Ohms 






Mai 


iganin 


__ 


\ 






100.03 






/ 








^ 










/ 








\ 




100.01 


s 




/ 








\ 


\ 


1 
















100.00 


















99.99 


/ 






Temp 


•rature 









10 



20 



40 



Fig. 3. 



60 



60 



70 



excess of zero; and at a definite temperature, which 
varies with different specimens, its coefficient is zero. 

The general character of the resistance-variations of 
manganin with temperature may be ascertained from the 
diagram (Fig. 3), in which temperatures are plotted as 
abscissas, and corresponding resistances of a hundred- 
ohm standard as ordinates.^ In this case the temperature 

1 Dr. Lindeck, Report of the Electrical Standards Committee of the British 
Association, 1892, p. 12; Proceedings of the International Electrical Congress^ 
1893, p. 165. 



BESISTANCE. 



25 



coefficient is positive up to 40° C, the absolute value, 
however, being very small, as the following table of the 
mean linear coefficients between the given temperatures 
shows : 

TABLE. 



Range 
of Temperature. 


Mean 

Linear Temperature 

Coefficient. 


Range 
of Temperature. 


Mean 

Linear Temperature 

Coeffi-cient. 


10° to 20° 
20° to 30° 
30° to 35° 
35° to 40° 
400 to 45o 


25 X 10 — 6 

14 X 10 — 6 

4x10-6 

3 X 10 — 6 

IX 10-6 


45° to 50° 
50° to 55° 
55° to 60° 
60° to 65° 


— 1 X 10 — 6 

— 2 X 10 — 6 

— 4 X 10 — 6 

— 5 X 10 — 6 



For most purposes the variability of the resistance of 
manganin with temperature may be quite neglected. At 
about 45° the resistance of the specimen under considera- 
tion passes its maximum, and the curve beyond this 
temperature shows a negative coefficient. 

23. Resistance Boxes. — The resistance of conduc- 
tors is commonly measured by comparison with other 
resistances the values of which are known with some 
precision. They are generally coils of insulated wire 
wound non-inductively on bobbins, and their values are 
so arranged that they can be used in any convenient 
combination. Collectively they make what is called a 
resistance box. 

Each bobbin is made non-inductive by the following 
method of bifilar winding : A length of wire sufficient to 
give more than the required resistance is cut off, bent 
double at its middle point, and wound double on its 
spool or form. This is done for the purpose of avoiding 
self-induction on starting or stopping the current. If 



26 



ELECTRICAL ME A S UBEMENTS. 



the coil is wound on a metal form, the form should 
be split longitudinally to prevent induction currents 
in it. The resistance of a length of wire is usually 
increased somewhat by bending as it is wound on its 
core. 

Each coil is exactly adjusted and finally fixed to the 
under side of the hard-rubber top of the resistance box. 

Its ends are soldered to two 
heavy brass or copper rods 
which extend through the 
hard rubber and are con- 
nected to massive brass 
blocks C\ C (Fig. 4), 
which offer no appreciable 
resistance. The coils are 
connected across the gap be- 
tAveen these blocks. When 
any brass plug P is with- 
drawn the current must pass 
through the coil bridging the gap between the discon- 
nected blocks. 

The coils are adjusted in ohms in series as follows : 
1, 2, 2, 5, 10, 10, 20, 50, 100, 100, 200, 500, and multiples 
of these. The total capacit}^ of the preceding series is 
1000 ohms. Or they may be arranged in this manner : 
1, 2, 2, 5, 10, 20, 20, 50, 100, 200, 200, 500, and so on, 
making an aggregate of 1,110 or 11,110 ohms. For a 
hundred thousand ohm-box there are commonly four 
coils, of 10,000, 20,000, 30,000, and 40,000 ohms, respec- 
tively. 

Resistance boxes are also made so that the coils may 
be joined in multiple. If coils of 25,000 ohms each are 
connected across from the block to 1, 1 to 2, 2 to 3, 




Fig. 4. 



BESISTANCE. 27 

and so forth (Fig. 5), they ma}^ be joined in multiple or 
in series by the plugs so as to give a resistance between 
the terminal binding-posts ranging from 2,500 to 250,000 
ohms. 

The plugs are slightly conical, and they should fit 
very exactly in the conical sockets reamed out between 
the ends of the adjacent brass blocks. Unless the fit is 
exact and the plugs are clean, the resistance of the con- 




tact will not be negligible, especially with coils of small 
value. The plugs should be kept very clean — free 
from dust, oxide, and grease. They may be cleaned by 
rubbing with a cloth dipped in a very weak solution of 
oxalic acid. In pressing the plugs into their places 
a firm pressure should be used while the plug is slightly 
turned ; but great care should be exercised not to seat 
them too rigidly or forcibly; otherwise their removal 
endangers their hard-rubber tops. 

Each resistance box is adjusted at some convenient 
temperature wlnqh should be marked on the box. Cor- 



28 ELECTRICAL MEASUREMENTS. 

rections may then be made to reduce to the resistance 
corresponding to the temperature of the box, which is 
ascertained at the time of use either by means of an 
attached thermometer, or by one passed through a hole 
provided for the purpose in the cover. 

The blocks to which the coils are attached should be 
pierced with a tapering hole for special plugs with bind- 
ing terminals, so that each coil may be put into the 
circuit separately for the purpose of comparing the 
resistances among themselves. 

It is very essential that a good resistance box be kept 
in an outer case to protect it from dust and the light 
when not in use. Direct sunlight on the hard-rubber 
top should be carefully avoided, since the sulphur in the 
rubber oxidizes in the light, especially in the presence of 
moisture, with the production of sulphuric acid. This 
greatly reduces the insulation of the hard rubber. 

24. Pohl's Commutator. — In the practice of many 
of the following methods of measurement, a commuta- 
tor for reversing the current 
through any portion of the 
circuit, or for switching from 
one circuit to another, is 
an indispensable appliance. 
PohFs commutator meets 
the purpose admirably. 

The six binding-p o s t s 
(Fig. 6) make connection 
with the corresponding mer- 
cury cups. The points e and / are connected with the 
source of the current. With the connecting wires ad^ 
ch^ in place, the apparatus is adapted to reverse the 




HESISTANCE. 



29 



direction of flow through the circuit connected with cd. 
In the position shown, e is connected with c, and / 
with d. But if the movable lever is tilted over, it is 
easy to see that e will be connected with d and / with 
e through the cross-connecting wires. Of course the 
two conductors at the ends 
of the tilting-switch are 
joined by an insulating stem 
of glass or hard rubber. If 
now the cross-conductors 
are removed, then when the 
switch is in the position 
shown, the points e and / 
are joined to c and d re- 
spectively ; but if the lever is 
thrown over, e and / are put 
in connection with another 
circuit from a round to h. 




VWVIAAA- 

X 

Fig. 7. 



25. Measurement of Re- 
sistance by Means of a 
Tangent Galvanometer.' — 
Connect the galvanometer, 

the resistance x to be measured, a battery of constant 
E.M.F., and a resistance box in series (Fig. 7). Then 
if 6 is the deflection and U the E.M.F. of the battery, 



U 



A tang 6 = 



B+ G--\- R + x " cot 6' 

In order to measure x by means of one observation 
only it would be necessary to know B^ the battery 
resistance, (7, the galvanometer resistance, E^ and the 
constant A. 



1 Foi' description of the tangent galvanometer^ see Article 62. 



30 ELECTBICAL MEASUREMENTS. 

But X may be determined without knowing any of the 
above quantities, as follows : 

Make two sets of observations mthout x, and with re- 
sistances Bi and Bo in the box, of such value that the two 
deflections 0^ and 6.^ sliall be respectively about 30° and 60°. 

Then 

^ =7l tang(9i, or ~ cote, = B+ G + B,; (1) 



B+ a + B, ^ A 

^^— ^-^ = ^tang ^„ or|cot 6., = B+ (7+ B,. (2) 

Subtract (2) from (1) and 

^ (cot d, -cot 6-?^ = B,-B,. ... (3) 

Then with x in circuit and a resistance B such that 
the deflection 6 may be intermediate between 6i and 62 , 
we have 

^ cote = B+ a+ x-^B (4) 

Subtract (2) from (4) and 

"^ (cot - cot 6.2) =x + B -B, ... (5) 

From (3) and (5) 

x-\- B — B2 _ cot 6— cot 6, 
B.-B. " (5ot^r=^^x^; 

and x = B.-B + CB,-B.f''^^^ -cot^.^ 

^ "^cot e, — cot 0.2 

Example. 

The tangent galvanometer gave the following deflections with 
the resistances indicated : 



Ohms. 




De 


FLECTION?;. 




COTAXGEXTS. 




Right. 




Left. 


Average. 




12 


31. o« 




31.0° 


31.0° 


1.632 


3 


62.5 




61. 


61.75 


0.537 


X 


44.7 




44.0 


44.6 


1.014 



BESISTANCE. 



31 



Therefore, 

x- = 3 + (12 - 3) 

In this case B was zero. 



1.014-0.537 
1.632 — 0.537 



= G 92 ohms. 




26. The Reflecting Gal- 
vanometer. — For the pur- 
pose of observmg a very 
small deflection of the 
needle of a galvanometer, 
a lioiit mirror is attached 
to the movable system, and 
a beam of light reflected 
from this serves as a long 
pointer without lo eight. 
Such a o-alvanometer of the 
" tripod " pattern is shown 
in Fig. 8 ; the mirror may 
be seen at the centre of the 
coil. The instrument is 
surmounted with a long 
rod, on which the curved 
magnet may slide up and 
down. It is held in place 
by friction. This magnet is 
employed to vary the sen- 
sitiveness of the instrument. 
To increase its deflection 
for a given small current, 
the plane of the mirror, 
which contains the mag- 
netic needle at its back in 
the form of several pieces 
of very thin watch-spring, is first made to coincide as 



Fig. 8. 



32 ELECTBICAL MEASUHEMENTS. 

nearly as possible with tlie magnetic meridian. The 
north-seeking pole of the control magnet is then turned 
toivard the north. It must be remembered that the 
magnetism of the northern hemisphere of the earth 
corresponds to that of a south-seeking pole ; that is, it 
produces at the needle of the galvanometer a magnetic 
held equivalent to that which would be produced by a 
permanent magnet with its south-seeking pole turned 
toward the north. Now, the object of the control 
magnet is to neutralize or compensate a part of this 
magnetic field if increased sensibility is desired. This 
it can do only when its north-seeking pole is turned 
toward the north. To make the sensibility a maximum, 
the magnet is slowly lowered; this lengthens the 
period of oscillation of the needle. If the control 
magnet is placed too low, it reverses the magnetic field 
at the needle, and the needle then turns completely 
around, with its south-seeking pole toward the north. 
The magnet must then be slowly withdrawn till the 
needle again returns to its normal position. The control 
magnet can be turned around slowly b}' means of the 
tangent screw on the top of the galvanometer. This is 
necessary for the purpose of placing the needle in the 
magnetic meridian after the control magnet is in position. 

27. The Multiplying Power 
of a Shunt. — Let g and s be the 
resistances of the galvanometer 
and shunt respectivel}^, measured 
between the two points A and B 
(Fig. 9) ; and let I^ and I, be the 
currents through the two paths. 
Let V be the potential difference (P.D.) between A and B. 




RESISTANCE: 



V V 

Then i = — ^ ai^cl Z = — . 

9 ^ 

Also if the total current is /, 

9 ^ 

But i' = -, and therefore ^i-=_^ = :^ 

Z ^ I, + Is s + g I 

Therefore 



s -4- ^ . 
The fraction ' ^-^ is called the "multiplying power of 

the shunt." It is the factor by which the current flow- 
ing through the galvanometer must be multiplied in 
order to find the total current. Also from the above 
equation 

If it is desired that Ig shall be yq of i, then 
= -— , or 10s = 5* + //, and g = 9s. 



s + g 10 
Whence s = ^g. 

If Ig is to be x-^o of Z then 

s = 2^9- 
If Ig is to be YoVo of /, then 
s 



These are the three relative values usually given to 
shunts in order to avoid inconvenient factors. Such 
shunts are applicable only to the galvanometers for 
which they are made. The plan of the top of such a 



34 



ELECTRICAL ME A S UBEMENTS. 



shunt-box is shown m Fig. 10. One end of all three 
coils is connected with the block A ; the other ends to 
the blocks (7, i>, E. The central 
block is connected to B. 

Shunts are also made for a current 
of tV. tw. and yoVo through the gal- 
vanometer, while the total resistance 
in the circuit remains constant. The 
entire current /thus remains the same 
Fig 10' whichever shunt is used. 




28. Two Methods of reading a Mirror Qalvanom- 
Q^QY — The deflection is read b}^ means of a scale of 
equal parts, preferably milli- 
metres, numbered contin- 
uously from one end to the 
other. Let BAB' (Fig. 11) 
be the scale, and let C be the 
mirror; and let the scale be 
so placed that it shall be par- 
allel to the galvanometer 
mirror when no current is 
passing. Then if the magnet 
and mirror have been turned 
through an angle 6, 
ACB = 2d, 
since the reflected ray of light is always turned through 
twice the angle of the deviation of the mirror. Also 

— = tang 261. 

The two methods of observing the distance AB are 
known as the '' lamp and scale " method and tlie " tele- 




BESISTAJSrCE. 



35 




Fig. 12. 



scope and scale " method. The device for lamp and 
scale is shown in Fig. 12. The, light of the lamp 
passes through an opening across which is stretched a 
fine wire corresponding to 
the point A of Fig. 11. 
After reflection from the 
mirror, the image of the 
wire falls on the diml}- illu- 
minated scale. In order to 
obtain a good image, a 
converging lens ma}' be 
placed some distance in 
front of the wire in such a 
position that the wire and the scale are conjugate foci 
for a beam reflected from the mirror, which in this case 
must be plane. But if a concave mirror, with a radius 

of curvature of 
about one metre, 
be used in the gal- 
vanometer, then 
the image of the 
wire will be focused 
on the scale when 
the wire is placed 
just below centre 
of curvature of the 
mirror. A translu- 
cent scale is much 
to be preferred. 
The observer is then on the side of the scale away from 
the galvanometer, and the reading is much more conven- 
ient. A gas-jet at one side may be used in place of the 
lamp ; and in this case a mirror at the back of the scale 




t^j^"* 



Fig. 13. 



36 ELECT PaCAL MEASUREMENTS. 

reflects the light through the opening containing the 
wire. 

In the other or subjective method of observing the 
deflection a telescope takes the place of the lamp and 
slit or wire. Such a reading telescope with attached 
scale is shown in Fig. 13. It is set up so that an image 
of the middle point of the scale is obtained by reflection 
from the galvanometer mirror when at rest with no cur- 
rent passing. If now the mirror is deflected the scale 
appears to swing across the field of view of the tele- 
scope, and when it comes to rest the observer reads the 
division of the scale coinciding with the vertical cross- 
wire in the eye-piece. Instead of the usual spider webs 
for cross-wires, fine quartz fibres may be substituted with 
most satisfactory results. If the galvanometer is to be 
used merely as a galvanoscope for detecting the passage 
of a current, then it is necessary only to observe whether 
the scale appears to move when the key is pressed. 

The telescope and scale possess the advantage that 
they can be used in a light room; and this method 
admits of greater accuracy than that of the lamp and 
scale, because the magnification of the telescope allows 
the divisions to be read to tenths. 

Let ??i and n., be the readings of the scale when no 
current is , passing and when deflected by a current 
respectively. Let a be the distance between the mirror 
and the scale and d the '' deflection." Then 
d — n.j — ni , 

and = - tan ~^ ~ . 



For small angles we may write approximately 

d_ 
2a 



e = tan e = sin e = —. 



EESISTANCE. 37 



H 8 = ~ the following . equations express the expansions 

of the several quantities in terms of the tangent of twice 
the angle : 



^=ljl- 


-i8= + ia^-i8«+. . 


1 


tan ^ = 1 j 1 - 


'¥^l''-i''^ ■ 


■\ 


sin e = 1 j 1 - 


-'f^l>-- ■ ■ 


■ ■ 


2-f = |jl- 


82 2048 





If the deflection does not exceed 6° the fii'st term of the 
correction is usually sufficient. 

Table I. in the Appendix gives the correction factors 
for the above four quantities from 8 = 0.01 to 0.2. 

Table II. gives the number to be subtracted from the 
deflection d to make it proportional to the tangent of 
the angle instead of the tangent of twice the angle, 
or to tan 6 instead of tan 20. 

29. Determination of the Figure of Merit of a G-al- 
vanometer. — The figure of merit of a galvanometer is 
the constant current which will produce a deflection 
of one scale division, or what is practically the same 
thing for small angular deflections, the ratio of the cur- 
rent to the deflection in scale divisions. If this ratio is 
not a constant for different values of the current, the 
galvanometer should be calibrated and the figure of merit 
calculated from the corrected readings. 

A convenient method of determining the figure of 
merit is to connect the galvanometer in series with a 



38 ELECTRICAL MEASUREMENTS. 

battery of known electromotive force -Z7, and a known 
resistance i?, which should be as large as possible and 
still give a suitable deflection. Note the deflection d of 
the galvanometer and calculate the current. For the 
latter it is necessary to know the resistance G- of the 
galvanometer and B of the batter}^ unless they are 
negligible in comparison with R If they are not neg- 
ligible and are unknown, they may be measured by 
means of methods described in articles 38 and bb. The 
figure of merit F is expressed by the following relation : 

F= A . 

In the case of a very sensitive galvanometer, it some- 
times happens that the deflection is excessive, even with 
the highest resistance at hand in series wdth the galva- 
nometer. 

In this case the galvanometer may be shunted by a 
coil of known resistance, preferably -J, gV, or q^q of that 
of the galvanometer. If the resistance of the galvanometer 

is n times that of the shunt, of the whole current 

n + 1 

passes through the galvanometer. The figure of merit 
is then expressed by the following relation : 

^-- <r~^ 

(B + ^^+^) (7^4-1) d 
n + 1 

As tiie deflection of the galvanometer depends on the 
distance of the scale from the mirror, it is customary to 
mention the distance at which the figure of merit is 
determined. The figure of merit of galvanometers carry- 
ing a compensating magnet may be varied between wide 



BESISTANCE. 



39 



limits by varying the strengtli of the magnetic field in 
which the suspended needle swings. 

30. Comparison of Resistances by Means of Poten- 
tial Differences. — Connect the unknown resistance x 
and a known resistance R of about the same value in 
series with a battery B of constant E.M.F. (Fig. l-i). 




Fig. 14. 

It may be necessary to use also another resistance r, 
which need not be known, but which may be necessary 
for the purpose of adjusting the current to the proper 
value, so as to secure a convenient deflection of the gal- 
vanometer. By means of a Pohl's commutator C^ the 
high resistance galvanometer Gr is connected first to the 
terminals of the known resistance i^, and then to those 
of a:, in such a Avay that the deflections shall be in the 
same direction. This operation should be repeated a 



40 



ELECTRICAL MEASUREMENTS. 



number of times till constant results are obtained. Then 
if c?i and ch are the deflections in the two cases, which 
should be as nearly as possible the same and not too 
large, we have 



or 



= R 



do 



The method proceeds on the assumption that the fall 
of potential is proportional to the resistance, and that 
the galvanometer deflections are proportional to the cur- 
rents flowing through the instrument, and therefore 
proportional to potential differences. 

Example. 

The followino: observations were made : 



ResiBtance. 


Reading. 


Zero Reading. 


Deflection 


0.3 


873 


500 


373 


X 


853 


500 


353 


Therefore 










0.3 X 353 


= 0.284 ohm. 





373 



P 'A^^ 



31. Measurement of Re- 
sistance by Means of the 
Diflferential Qalvanometer. 
— A differential galvanom- 
eter is wound with two coils 
of approximately equal resist- 
ance and equal magnetic field 
at the centre of the coils. 
The connections are made, as 
shown in the diagram (Fig. 
15), the two parts into which 
the current divides going in 



BESISTANCE. 



41 



opposite directions round the two coils. The observa- 
tions consist in adjusting the resistance M until the gal- 
vanometer shows no deflection on closing the circuit. 
In case an exact balance cannot be obtained, the fraction 
of the smallest division of B, usually one ohm, necessary 
to produce a balance, can be determined by means of 
deflections in both directions and interpolating. If d^ 
is the deflection mth B ohms, and ch the opposite 
deflection with B + 1 ohms, then the resistance to 
balance is 

Cli + ^2 

It is essential to determine whether the two coils are 
of equal, resistance, and whether the same current 
through each produces the same 
magnetic field at the centre. For 
this purpose connect the two coils 
in series, but so that they shall 
produce opposing magnetic fields 
at the needle. If the needle shows 
no deflection, the coils are balanced 
magnetically. If there is a deflec- 
tion, a balance may be secured if 
one coil is movable, as in the 
Eclelmann galvanometer, by vary- 
ing its distance from the needle ; 
or it may be secured by passing 
one-half of the current through a 

coil properly placed under the galvanometer, or in its base. 
Such an adjustment, however, is usually troublesome. 

A much better method is the following : If necessary 
insert a resistance r in one branch, as shown in the dia- 
gram (Fig. 16), in order to effect a balance. This 



p 

I 



Fig. 16. 



42 ELECTRICAL MEASUREMENTS. 

resistance may be simply a small increase in one of the 
lead wires, or it may be a good many ohms. It is 
advisable to introduce a resistance P in the battery 
branch to diminish the current. Let A and B be the 
resistances of the two windings, including the connecting 
wires and resistance r, between the points of division of 
the circuit. Then let the resistances R and x be inserted 
as in Fig. 15, and let a balance be obtained by deflections 
in the two directions and by interpolation if necessary. 
Next exchange R and x and balance again. Let Ri and 
R2 be the resistances to balance in the two cases. 
Then A : B : : R^ : x, for the first balance, 
and A : B : : x : R., for the second balance. 



Whence x=^\/R^.R,. 

Example. 

I. To determine the resistance of one B.A. unit in ohms : 
Apparatus. — Edelmann's mirror galvanometer with high resist- 
ance coils. 

A B.A. unit box for the unknown resistance {x). 

An international olim box for known resistance {B). 
Cond. I. — The influence of both coils traversed by the same cur- 
rent, but in opposite direction, should be equal for a magnetic 
balance. 

Current through A alone deflects to smaller numbers. 

Current through B alone deflects to larger numbers. 

Current through both coils deflects to larger numbers. 

B was moved 4.-5 mm. away from the needle ; then there was 
no deflection. 
Cond. II. — Resistance of both coils should be equal for elec- 
trical balance. 

Current flowing through both coils in parallel deflects to larger 
numbers. 

Resistance put in series with B until no deflection was observed. 

Resistances x and R inserted. 

a; =1,000 . . . i? = 986; no deflection. 



RESISTANCE. 43 

The galvanometer was not sensitive enough to estimate R to 
tenths. 

Hence a: = 0.986i?, 

or one B.A. nnit equals 0.986 of an international ohm. 

II. To determine the resistance of one B.A. unit, in ohms, by 
the second method : 
Apparatus. — A Thomson astatic mirror galvanometer. 

Kesistance of B.A. box, right at 16° C, as unknown resist- 
ance {x). 

Resistance box, in ohms, right at 17° C, as known resist- 
ance (i?). 
Formula : x =^ \^ Bi • B2 - 

Adjustment of Aptparatus : 

A current through coil A deflects to smaller numbers. 

A current through coil B deflects to larger numbers. 

A current through both coils deflects to larger numbers. 

In order to get no deflection 1170 ohms (r) were added to coil 
B, with A and B in parallel. 
Observations : 

First. X in series with A ; R with B (R does not include the 
1170 ohms). 

(a) x' =500 B.A. . . . i? = 363.1; no deflection. 

(b) x" =600 B.A. . . . i? = 435.65 ; no deflection. 

(c) a;'" =800 B.A. . . . 7? = 580.71 ; no deflection. 
Second. Resistances x and R exchanged. 

(d) x' =500 B.A. . . . R= 671.7; no deflection. 

(e) 2;' =600 B.A. . . . R= 804.83; no deflection. 
(/•) x'" = 800 B.A. . . . R = 1072.83 ; no deflection. 

Calculation : 

From (a) and {d),x = ^363.1 X 671.7 = 

493.86; — = 0.98772. 
x' 



From (6) and (e), x = v'^35. 65 X 804.83 



592.14; ^ =0.98690. 
x" 



From (c) and (/), x = V'580.71 X 1072.83 = 



789.29; — = 0.98661. 
' ..III 



Mean 0.98708. 



44 



ELECTRICAL MEASUREMENTS. 



Correction for temperature : 

Temperature of both boxes, 20.5° C. Temperature coefficient 
for both, 0.00044. 

1 B.A. unit at 20.5° C. = 1 + (0.00044 X 4.5) = 1.00198. 

1 ohm unit at 20.5° C. = 1 -f- (0.00044 X 3.5) = 1.00154. 

Therefore 1.00198 B.A. units = a; X 1.00154 ohms. 



Whence 1 B.A. unit 



1.00154 
0)0198 



X 0.98708 = 0.98664 ohm. 



32. Heaviside's Modification of the Differential 
Galvanometer.^ — Instead of dividing the current from 
the battery between the two coils, join the coils so that 
the same current passes through both of them, and by 

reversing one of the coils g' 
(Fig. 17), prevent the current 
from influencing the needle. 
The rheostat B is connected 
in parallel with one coil g and 
the resistance x to be meas- 
ured in parallel with the other 
(/. When M equals x it is 
easily seen that the currents 
in g and g' are equal provided 
g and g' are equal to each 
other. But this method may 
be used exactly as in the last 
article. Let B^ be the resistance to balance x in the 
relative positions shown in the figure. Then exchange 
the rheostat and the unknown resistance and balance 
again, interpolating, if necessary, 
ance in the rheostat. Then 

Hi : X : : g : 
and X : R, 




and let R., be the resist- 



9 



9'- 



Whence 



X = \/it, • R^ 



^ Electrical Papers, Vol. I., p. 13. 



RESISTANCE. 



45 



This method assumes that the galvanometer is magneti- 
cally balanced. If the galvanometer is not magnetically 
balanced, the stronger coil may be shunted with a resist- 
ance r (Fig. IT), such that when the two galvanometer 
coils (one shunted and the other not) are placed in 
series, no deflection is obtained. When x is greater than 
g the other method is to be preferred. But for values oi 
X less than </, the present method gives greater sensi- 
bility. If, for instance, the battery have a resistance of 
10 ohms, each coil of the galvanometer 500 ohms, and x 
is 10 ohms, then the Heaviside method is seven times as 
sensitive as the first method. 

33. Wheatstone's Bridge. — Wheatstone's Bridge 
is a combination of resistances most commonly employed 
to measure all except a very high resistance or a very 
low one. It consists of six conductors connecting four 
points, in one of which is a source of electromotive 
force, which need not be constant ; and another branch 
contains a galvanometer. 

Let AJB CD (Fig. 18) be 
the four points connected by 
six conductors. Then since 
the fall of potential by the 
two paths between A and D 
is the same, there must be 
a point B on the path ABD 
which has the same poten- 
tial as another point on the 
path A CD. If these points 
are joined by a conductor, 

including a galvanometer, no current will flow through 
it, and we have the relation 




46 



ELECTRICAL MEASUBEMENTS. 



For let Ji be the current through Ri. It will also 
be the current through ^3, since none flows across 
through the galvanometer. Also let I2 be the current 
through the other branch A CD. Then since the poten- 
tial difference between A and B is the same as between 
A and C, 

B.,I, = RJ, (1) 

Similarly, BJ, = RJ, (2) 

Dividing (1) by (2), f =f . 

This may also be written, 

R\ R^ 

ErR,' 

or Ri : Ro : : R^ : R^ . 

The last equation might have been obtained by bal- 
ancing with the galvanometer connecting AD and the 

battery applied to the points 
BC. The conditions for a 
balance are, therefore, the 
same after the galvanom- 
eter and battery have ex- 
changed places as before, 
and depend only upon the 
proportionality of the four 
resistances. 

If the six conductors are 
arranged as shown in Fig. 
19, and if 




Fig. 19. 



^1 : R2 : : Rz : R^ , 

so that no current flows through the galvanometer, then 
any change of E.M.F. in AD will not produce a potential 



BESISTANCK 



47 



difference bet^Yeen B and C; the converse is, therefore, 
true, so that the battery and galvanometer may exchange 
places without disturbing the balance. The balance is 
in no way dependent upon the resistance of BC and AD, 
though the sensibility of the arrangement is dependent 
upon these relative resistances. AI) and BO are said 
to be conjugate; that is, they are connected by this 
mutual relation of independence. So, also, when the 




6 0^ 060 [ 




o 



o 



o 



1 ^ OoH 



Fig. 20. 

corresponding resistances are proportional, AB and BO, 
BB and AO are conjugate. 

To use the Wheatstone's bridge for the measurement 
of a resistance, three known resistances are taken, having 
such a relation to the unknown x that a balance is 
obtained with the galvanometer. In practice two resist- 
ances, Ri and R2 , are chosen, and B^ is made t(^ vary 
till a balance is secured. Then 



X — Ra 



b: 



48 ELECTRICAL MEASUREMENTS. 

Maxwell gives the following rule for the connection 
of the battery and the galvanometer to the four resist- 
ances : ^ "Of the two resistances — that of the battery 
and that of the galvanometer — connect the greater 
resistance so as to join the two greatest to the two least 
of the four other resistances." 

If, for example, 

R, = 1000, R, ^ 10, R, = 3752, x = 3T.52, 

then the battery should join the point between the two 
proportional coils to the junction of R^ and x^ as shown 
in the diagram (Fig. 20), if the resistance of the galva- 
nometer is greater than that of the battery, whicli is 
usually the case. 

The battery circuit sliould be closed first, and then the 
galvanometer circuit, so as to avoid the effect of any 
self-induction in the coils of the resistances. A double 
successive contact key is very convenient for this pur- 
pose. It opens the two circuits in the inverse order to 
that in which they are closed. 

34. The Post-Office Resistance Box. — One of the 
most convenient arrangements for the use of the Wheat- 
stone's bridge method is the Rost- Office Resistance Rox, 
so called because of its employment in the telegraph 
department of the British post-office. Fig. 21 is a plan 
of the top of this box. 

The arms AR and A consist of two sets of propor- 
tional coils — two lO's, two lOO's, and two lOOO's. Any 
pair of these represent the resistances Ri and Ro of Fig. 
18, which is lettered to correspond with the plan of the 

^ Electricity and Magnetism, Vol. I., p. 438. 



RESISTANCE. 



49 



post-office box. These proportional coils may contain, 
also, a pair of I's or a pair of 10,000's 



The ratio, — ^ is 
M2 



then either 1, 10, 100, 1000 or 1, ^\, y^o, loVo- The 
unknown resistance may be measured directly to y^^ or 
10^00 of the smallest coil included in the rheostat arm 




50 100 10 10 100 IC 

. 2 3 I i 1 



^:3C3CJC3 




300 200 



GX X Inf 



D400 lOCO 
■OOCJ 








40 ^ ^ 30 



(3r3c:^(3 



3000 4000 





Fig. 21. 



UFaB. Thus, if B, is 1000, E, 10, and E, 253 ohms, 
then X is ywo of 253 or 2.53 ohms. No binding-post is 
provided at A^ but A is joined by a wire under the hard- 
rubber top of the box to a stud at a, so that it is put in 
connection with A^ by pressing the key A^a. In the same 
way the terminal B^ is put in connection with B by 
pressing the key B^b. Connection is made between B 
and B by a heavy copper strap not shown. This is 



50 



ELECTRICAL ME AS UREMENTS. 



screwed down tightly by the binding-screws B and E. 
Since two wires must be connected at both C and D, 
these points are provided with double binding-posts. 
At the point marked Inf, is the infinity plug. When 
this plug is out, the circuit through the rheostat arm is 
completely broken. It will be observed that the series of 
resistances shown are 1, 2, 3, 4, 10, and multiples of these. 
If it is impossible to obtain a balance with the smallest 
coil in the rheostat arm, then the fraction required to 
balance may be determined by observing the deflections 
of the galvanometer, first in one direction and then in 
the other, and the true value of x ma}- be found by 
interpolation. For example, let the following be the 
resistances and deflections in divisions of the scale : 



^3. 

1206 
1205 



Deflections. 
Left. Right. 
6 
U 



Then one ohm causes a change in the deflection of 
20 divisions. Hence the value of i?3, which would give 

14 



an exact balance, is 1205 



20 



, or 1205.7. 







Example. 


Ratio of proportional 


coils 7?i and Ro, 1000 : 1000. Galvanom- 


eter used with ^-^^^ 


shunt. 


Rz 




Deflection. 


ohms. 




To liigher numbers. 


100 




" lower " 


40 




" " 


20 




" " 


10 




(( a u 


4 




" higher " 


6 




" lower *' 


5 




Almost none, slightly to lower numbers. 



RESISTANCE. 51 

Changing the ratio of Ri and ^2 to 1000 : 10 and removing the 
galvanometer shunt, the following observations were obtained : 

7?3 Deflection. 

500 To lower numbers. 

495 " higher " 

497 10 mm. to higher numbers. 

498 33 mm. to lower " 

Therefore to give no deflection R3 should be 497i§ = 497.23, or 
X = 4.9723. 

From this must be subtracted the resistance of the lead wires, 
which was obtained as follows : 
Ratio of Ri and R2, 1000 : 10 : 

^3 Deflection. 

8 To lower numbers. 

1 75 mm. to higher numbers. 

2 16 mm. to lower " 

Therefore to give no deflection R3 should be l|f = 1.82, or the 
resistance of the lead wires was 0.0182, giving for the resistance 
of a;, 4.9723 — 0.0182 = 4.9541 ohms. 

The temperature of the box was 20° C. ; and as it was right at 
17° C. and had 0.00023 for its temperature coefiicient, the final 
corrected value for x was x = 4.9541 [1 ■+- 0.00023 (20 — 17)] = 
4.9576 ohms at 20° C. 

35. The Slide Wire Bridge. — -Since it is necessary 
to know only the ratio of Ri to Hj, and not their abso- 
lute values, the resistances of two adjacent portions of a 
uniform wire may be employed in place of adjusted 
coils. 

With the openings at 1 and 2 (Fig. 22) closed by 
heavy copper straps, obtain a balance by moving the 
contact C along the wire. Then 

X a r, a 



52 



ELECTRIC A L ME A S UREMENTS. 



The resistance of the two parts of the wire a and h are 
here supposed to be proportional to their lengths. 

A single determination of a resistance by this method 
does not admit of very great exactness, since the position 
of C may not be read with precision, and the wire may 
not be of the same resistance for each unit of length. 




Fig. 22. 



36. Effect of Errors of Observation. — An error in 
reading the position of C produces the smallest effect on 
the result when C is at the middle point of the wire. 
This may be demonstrated as follows : We have from 
the preceding 

a 



c 



(1) 



when c is the entire length of the wire 



RESISTANCE. 53 

Suppose now an error./ has been made in reading the 
position of the contact C on the bridge wire. Then 
the value of 2: is a; + F^ in which 

x + F=B^^±l- (2) 

c — a — / 

The general formula to apply in determining the con- 
ditions for the least error may be derived as follows : 

Let X be the observed quantity. 

Let X be the derived quantity. 

Also let/ be the error in the observed quantity, and 
let F be the resulting error in X. 

The error F arises from the use of x +/ instead of x in 
the equation connecting x and X. Then the relation 
of the four quantities is expressed by the equation 

F=/J^. ...... (8) 

F and X are quantities of the same kind ; also / and 

X. The partial differential coefficient — expresses the 

6x 

rate of variation of X with respect to x^ other variables 

for the time being considered constants. This rate, 

multiplied by the error in the observation, gives the 

total error in the result, or F. 

Applying this formula to the present case, we have 

from (1) 

bX bx T> (^ X ( N 

bx ba (c — «)■ 

since a is the observed quantity and x tlie derive 1 resist- 
ance. 
Whence 

F = fB^~^., and - = /" . . (5) 
((? — ay X a (c — a) 



54 ELECTBICAL MEASUREMENTS. 

This ratio will be a minimum when a (c— a) is a 

maximum. But the product of two quantities whose 

sum is a constant (c) is a maximum when they are 

equal to each other, or when a=c — a. In that case 

c 
'2a= c OT a= ^; or the contact C is at the middle point 

of the wire. H and x should therefore be made as 
nearly equal as possible. 

37. Use of the Slide Wire Bridge— First Method.' 
— Referring to the figui^e of Art. 35, it will be seen that 
the resistance of the copper bars, straps, and contacts 
from iVto X and from iV^ to M are measured in with a 
and h respectively. It may further happen that the 
index line of the slide is not exactly over the metal 
edge making contact with the bridge wire. Let / be 
this error, so that the true bridge reading is «i + /. Let 
Ti be the resistance of the bridge between iV and rr, and 
To that between iV' and M. It is necessary to observe 
that 7\ and r., are here expressed in terms of the resistance 
of unit length of the bridge wire. Then 

B 1000- («! + /) + ^2 • • • ^^^ 

if the bridge wire is divided into 1000 parts. 

Let now the positions of x and R be reversed. Then 

a;^ 1000-(^. + /) + r. 

where a. is the new bridge reading to balance. 
Adding numerators and denominators, we have 
x_100 + r, + r2-\- (ja^—a-?) 
lf~ 1000 + 7\ + n — {a, - a.) * 

1 Stewart and Gee's Practical Physics, Part II., p. 148. 



(2) 



(3) 



BESI^TANCE. 55 

The error f is thus eliminated. Moreover, the equation 
contains the small quantity Vi + r^ added to a large num- 
ber in both numerator and denominator. 

If the resistances Vi and ra are disregarded, then the 
formula becomes 

X _ 1000 + (^1 — ^a) ^A^ 

R~ 1000 -(a, -a.?}' ' ' ' ^ ^ 

If we consider formula (3), it will be evident that r, + rg 
would make no difference in the ratio if x and H were 
equal to each other, for their addition to numerator and 
denominator would be the addition of equals to equals, 
the ratio remaining unity. But under these circum- 
stances «! — a.j equals zero ; and the larger the numerical 
value of «i — (22, the greater will be the error introduced 
by neglecting the resistance ri + ^2 . Hence R should 
be adjusted so as to be as nearly equal to x as possible. 

Example. 

It was desired to determine the resistance of a coil marked 1000 
B.A. units. 1000 ohms in a box made by Nalder Bros, was used 
as the known resistance. 

Reading on the bridge wire 497 

Reading after exchanging x and E 505 

Here «i — a2 = — 8, 

and ^_ 1000-8 

or x = 984.1 ohms. 

The temperature of the boxes was 23° and the known resistance 
was right at 15°. Its temperature coefficient was 0.00044 ; there- 
fore the corrected value of x was 

X = 984.1 [1 4- 0.00044 (23 — 15)] 
= 987.6 at 23°. 



56 



ELECTEICAL IMEASCREMENTS. 



38. Galvanometer Resistance by Thomson's Method. 
— Connect the galvanometer, whose resistance is to be 
measured, in one of the proportional branches, AB, of a 
Wheatstone's bridge (Fig. 23). A second branch, BC, 
should consist of a resistance B-^, as nearly equal to the 
resistance of the galvanometer as convenient. The other 
two proportional branches, Bi and i?j, are obtained on 




Fig. 23. 

the Avire of a slide metre bridge. The battery branch, 
which shoukl be made up of a Daniell or other closed 
circuit cell, a resistance ?•, and a plug-key Ki should 
join A and C. The last branch should consist of wires 
of low resistance and a key Ko. 

We should now close the plug-key Ki in the battery 
branch and adjust the resistance r until the galvanometer 
gives a large steady deflection. If the deflection goes 
beyond the end of the scale, the scale may be moved 
until a reading is obtained. The actual value of the 



BESISTAJVCE. 57 

■reading is not- important. So long as ^, remains open 
there slionld be no change in the deflection, no matter 
where on the slide wire the point D may be taken ; and 
if a point on this Avire is found at which the potential is 
the same as that at B^ key K., may be closed and there 
Avill still be no change in the deflection. In this case 

^1 : M2 : '. Jls : Gr. 

If a slide wire bridge or its equivalent is not obtain- 
able, two resistance boxes may be used for E^ and M, . 
It will be foimd most convenient to keep the sum of 
their resistances constant, otherwise there will be different 
galvanometer readings with each different value of their 
sum, even before K^ is closed. 

For galvanometers of the d' Arson val type (Art. 70) 
the slide wire of low resistance is much more convenient 
than the resistance boxes, as it acts like a low resistance 
shunt to bring the galvanometer to rest ; however, with 
the resistance boxes a shunt of low resistance may be 
used in addition, which will practically accomplish the 
same thing. 

Instead of one cell of battery and a resistance r, we 
may use two cells of slightly different E.M.F.'s in oppo- 
sition to each other. Their difference will in general 
give sufficient E.M.F. 

It is not well to exchange the battery and the key Ky , 
although a balance may be obtained in this Avay; for 
each change in the position of D would then give a dif- 
ferent galvanometer reading, which would make the 
experiment very tedious, as it would be necessary to 
wait for the galvanometer to come to rest after each 
change in the ratio. 

It is necessary in this, as in other experiments with 



58 ELECTRICAL MEASUREMENTS. 

the slide wire bridge, to exchange the positions of G- and 
i^g and find the new position of I) to give a balance. It 
is also advisable to have a commutator in the circuit to 
reverse the direction of the current, although errors due 
to differences of temperature are practically eliminated 
by exchanging Gr and Jl:^. 

In the practice of this method it will be found con- 
venient to make a trial measurement of G- with any 
convenient value for i^., and determine the value of G 
roughly. For this it is not necessary to exchange G and 
i^o . Next make M^ as near the value of G as convenient, 
say to the nearest ohm ; then proceed as above to make 
the more exact determination. The reason for making 
i^3 as nearly equal to G as possible is that the resultant 
error is a minimum when D is at the middle of the slide 
wire. 

Example. 

First, i?3 = 100 ohms ; i?i = 611.4 ; i?2 = 388.6 ; .-. G = C3.56 
ohms. 

Second, make R3 = 64 ohms. Then i?i = 503.2 ; R^ = 496.8. 

Exchanging Rs and G, Ri = 498.8 ; R2 = 501.2. 

Therefore (/ = 64 i2^2_Zl4^= 63.44 ohms. 
1000 -\- 4.4 

In both cases changing the direction of the current had no 
effect on the values of the readings. 

39. Use of Slide "Wire Bridge — Second Method. 
— The bridge can be made more sensitive by inserting 
two resistances, Mi , R.2 , in the openings at 1 and 2 (Fig. 
24). These resistances should also be nearly equal to 
each other, or, more strictly, should have the same ratio 
as X and M. If the resistance of unit length of the 
bridge wire is p, and a and h are the two parts of the 



BESISTANCE. 



59 



wire on either side of the slide when a balance has been 

secured, then 

X _ Bi -^ ap 

B~~ B2+ ho' 

The value of x is thus known if p has been determined. 
Since the resistance of a and h now form only a small 
part of the total resistance of their respective branches, 
any error in reading the position of the slider must pro- 




Fig. 24. 

duce a smaller effect in the resulting value of x than 
when J?i and R2 are not used. These auxiliary resist- 
ances may be considered simply as extensions of the two 
ends of the bridge wire. 

If we introduce n and r^ as before, and suppose R^ and 
R2 expressed in terms of a division of the bridge wire, then 

X _ i^i + r, 4- «i /|x 



Reversing, 



R R2 + r2+ c— a 

X _ R2+ r., -\- c — a 
li ~ ~jRi + ri + a. 



(2) 



60 ELECTRICAL MEASUREMENTS. 

Here c represents the entire length of the wire. 
Adding (1) and (2) by addition of numerators and 
denominators, 

E~~ Bi^ E2-hri +7-2 +c— («! — as) * ' ' ^ ^ 
Put Ei-{- B2 + Ti + To + c = r, and ai — a2 = d. Then 

i? r-7/' • • • • • W 

If d is small compared to r, Ave may neglect small 
quantities of the second order aiid write, 

l-'-T (^) 

If the bridge is a metre long divided into millimetres, 
then the greatest value that d can have is 1000, and the 
least may be perhaps .2 mm. 

Let r = 5000 ; then from (4) 

a; _ 5000 + 1000 _ 3 
E 5000-1000" 2* 

This gives the maximum ratio of x to E to which the 
method is applicable. 

From (6) |= 1 + |^^ = 1.00008. 

This is the smallest ratio of ic to i^ for which the 
bridge can be used with the assumed extensions, E^ 
and Eo^ each resistance twice that of the bridge wire. 

The effect of increasing r is to make the ratio of the 
resistance of the bridge wire to the whole resistance of 
the wire and extensions or auxiliary resistances, Ei and 
jKs? smaller; this reduces the range of the bridge. 



BEtilSTANCE. 



61 



Fig. 25 is a bridge in which the connections are con- 
veniently arranged to exchange x and B by means of 




Fig. 25. 

a single commutator. Fig. 24 shows the connections 
with end resistances attached. The contact maker is 
carried on a long brass rod by means of a sleeve, which 
can be clamped at any point, and the final adjustment is 
made by means of the attached slow-motion screw. The 
scale is divided into millimetres, and a vernier reads to 
tenths. Fig. 26 is a section of the contact device 




Fig. 26. 



designed to allow a pressure on the bridge wire not 
exceeding a limited amount, which is governed by the 
small spring above the inner piston T\ The button /iT 
is depressed against the force of the larger outer spiral 
spring. The descent of T carries T with it till contact 



62 ELECTRICAL MEASUBEMENTS. 

is made with the wire. The piston T^ then enters T 
against the pressure exerted by the small spiral. This 
de^dee prevents any injury to the bridge wire by 
careless and excessive pressure on K. S is the sleeve 
which slides on the long brass rod, Sc is the scale, and 
T^the vernier. A short piece of the wire used on the 
bridge is soldered to the bottom of T, so as to make 
contact on the bridge wire at right angles. The rod T 
is prevented from turning by a square shoulder at the 
top where it passes through the outer housing which 
encloses the larger spring. This device, made by our 
mechanician, R. H. Miller, has proved very satisfactory. 

Example. 

Apparatus : New bridge (least reading 0.1 mm.). 

To measm'e resistance of manganin coil in oil. 

Two nearly equal resistances of about 5 ohms used for length- 
ening the bridge wire — Bi and i?2 • 
A. Observation I. : R on side with i?2, and x with Ri . 



Then 



X /?i H- ai 

R i?2 -|- c — ai 

R IN Ohms. Reading of Bridge. 

4.6 55-1.0 

4.7 275.6 

Observation II. : R and x exchanged. 
Then X R.-^c-a,^ 

R IN Ohms. Reading of Bridge. 

4.6 463.3 

4.7 739.5 

Determination of Ri (Art. 40) : 

2000 ohms 40 Ri -h 362.2 



(a) 



50 ohms 1 637 

.-. i?i= 25149.4. 



(6) 



RESISTANCE. 
2000 ohms _ 50 _ Z?i -}- 487.1 



40 ohms 1 512.9 

.-. i?i = 25157.9. 
Mean value for i?i = 25153.6 parts of the bridge wire. 
Determination of Bo : 

(a) 2000 ohms ^ 40 ^ Z?2 + 367.2 

50 ohros ~~ 1 632.8 ' 

.-. i?2 = 24944.8. 

(b) 2000 ohms __ 50 _ i?. + 492.0 

^ 40 ohms ~'T~ ~508 " 

.-. i?2 = 24908. 

Mean value of B2 = 24:926 A parts of the bridge wire. 

Calculation : 

Formula, ^ = 1+1, 

R r — d 

in which r := i?i -|- i?2 + c, and d = ai — «2 . 

Therefore, r = 25753.6 + 24926.4+ 1000 =51080, 

and rf = 90.7 for jR = 4.6, and —463.9 for i? = 4.7 ohms. 

TT X 51080 + 90.7 , r,r.op.P 

Hence, - — = ^ = 1 . 00356 ; 

4.6 51080 — 90.7 

and ic = 4.6 X 1.000356 =4.615 ohms. 

Also, _^_ 51080 -463.9 ^^3 

4.7 51080 + 463.9 

and X = 4.7 X .982 = 4.615 ohms. 

40. To find Ri and R2 in Terms of the Divisions 
of the Bridge Wire. — If the auxiliary resistances ^1 
and i^2 are used, the resistances Vi and n witli a good 
bridge will be small in comparison, and they may safely 
be disregarded. Close the opening 2 with the heavy 
copper strap provided for the purpose, and put Mi in the 
opening 1. Then with two known resistances, P and 



64 



ELECTRICAL MEASUREMENTS. 



Q (Fig. 27), obtain a balance and let the bridge reading 
be a. It is evident that P should be larger than §, or 
the point on which a balance may be obtained may lie 




Fig. 27. 

beyond the limits of the actual wire of the bridge, since 
7^1 is an extension of this wire. Then 
P Bi+ a 



B. = ^Co-a) 



a. 



Q c-a 
Whence 

R.2 may be determined in the same way. 
Example. 

^ = 5, a =304. 



Then 



i^i = 5 (1000 — 304) — 304 
= 3176. 
This result should be checked by measuriDg the resistance of 
the bridge wire and i?i independently. 

41. The Carey Foster Method of comparing" Re- 
sistances.^ — This method is especially useful for the 

1 Philosophical Magazine, ^Nlay, 1884 ; Glazebrook and Shaw's Practical 
Physics, 2cl ed., p. 561. 



HESISTAlv^CE. 



65 



piu'pose of determining the difference between two nearly 
equal resistances of from one to ten ohms. The method 
is as follows : 

Let aS^i and aS'. (Fig. 28) be the two nearly equal 
resistances to be compared, and let li^ and B., be two 
neai-ly equal auxiliary resistances, which should not 
differ much from aS\ and S-, . Let n and Vo be the resist- 



K,. 



R, 



L=4 



N ^^ 



'I'M. I... ■!.... I.. ..I....! 



-f 



,l,,.,i,..,l, 



,l,,,.i,,,,l 



Fie. 28. 



ances of iVilf and iV'iltf' respectively. Then if p be the 
resistance of unit length of the bridge wire, 

Bi_ S^ + n + pai .-j^x 

R, S, + r,^ ph 

Let now Sx and So exchange places, and let a., be the 
reading on the bridge wire for the new balance. 
Then B.^S^±n±^ ^2) 

Adding unity to both sides of (1) and (2), we have 
Rj + Ro ^ aS\ + )\ + p a , -i- jS, + n + jQ^i ^ 
Ro S, + r, + pb, 

S2 + n + p a.2 + Si + r. + ph, ^ 

Si+r> + ph, 



66 ELECTRICAL MEASUEEMENTS. 

Since ai -\- bi = c = ao + b.,^ the numerators of these 
fractions are equal; hence the denominators are also 
equal, or 

Si + r.2 + pb. = S. + r, + pbi . 

Therefore Si — S., — p (bi — b.?} = p («,. — cii) . 

The difference in the resistance of the two coils, Si and 
S-2^ is therefore equal to the resistance of that part of the 
bridge wire between the points at which the slide rests 
for the balance in the two positions of the coils S^ and S.. . 

Example. 

Si = coil No. 273, 0.99795 of an ohm at 15.4° C. Temperature 
coefficient 0.00023. 

S.2 = coil No. 194. Resistance to be determined. 

p = 0.00095459 at 20° C. (Art. 42). 

^1 left, Si right, reading 508.1 ? Temperature of Sx and S^ 

S-2 left, ;Si right, reading 497.25 S 19.3° C, of bridge 20° C. 

.-.5^2 = 0.99795 [1-h .00023 (19.3— 15.4)] — 0.00095459 (508.1 
— 497.25). 

;S2 = 0.98849 ohm at 19.3° C. 

42. The Determination of p. — The methods to be 
pursued in the determination of the resistance of unit 
length of the bridge wire will depend to a considerable 
extent upon the value of this resistance and the length 
of the wire. Since 

Si — So = p Ccio — ai\ p == — . 

do — «i 

Hence, if the difference between the resistance of the 
two coils Si and So is kno^vn, p can be found by deter- 
mining by two successive balances the length of the 
bridge wire corresponding to this known difference. 
For this purpose three standard coils may be used, two 



BESISTANCE. 67 

1-olim coils and one 10-ohm. The 10-ohm coil and one 
of the units are placed in multiple on one side, and the 
other unit on the other. The resistance S2 of the two in 
parallel is 

1 X 10 _ 10 

1+10 11* 

Hence S,-S. = l -^ =X = .09091, 



and 



P = 



11 11 

.09091 



If the entire resistance of the bridge wire is consider- 
ably in excess of one ohm, then p may be found by the 
aid of a single standard ohm and a heavy copper link, 
the resistance of which may be neglected. Then 

1 
p = . 

ao — ax 

With 1 and 100 ohms in parallel the difference between 
1 and the two others in parallel is .009901. 

A third method may be used when only one standard 
coil (and that of greater resistance than the bridge wire) 
is available. In the particular case considered the bridge 
wire really had a resistance of about 20 ohms ; but, to 
obtain greater sensitiveness, it was used with a coil of 1 
ohm resistance in shunt. The equivalent resistance of 
the combination was then about f f of an ohm, and the 
difference of readings on the bridge wire was increased 
about twenty times. The standard coil used, marked 
"No. 273 — 1 'legal' ohm at 12.8° C," called coil A in 
what follows, had a resistance of 0.99795 of an ohm at 
15.4° C. The two other coils were taken as unknown 
quantities. Coil B was a standard coil marked "No. 



68 ELECTBICAL MEASUREMENTS. 

194 — 1 B.A. unit at 15° C." This value, however, was 
someAvhat in error. The third coil C was of about f of 
an ohm resistance. By making the resistance of (7 a 
mean between that of A and of A and B in parallel, the 
effect of errors of observation was reduced to a minimum. 

In the first arrangement coils A and B were placed on 
opposite sides of the bridge, and their difference meas^ 
ured in terms of p. In this, as in the following arrange- 
ments, the coils were in water baths of practically the 
same temperature as that of the room. It is necessary 
for this experiment that A should be of exactly the same 
temperature as B^ though that of C may be different. 

To obtain this equality of temperature the water in the 
two water-baths should be well mixed, repeating the oper- 
ation several times if need be. If the coils and the 
water are practically at the temperature of the room, the 
whole will rapidly reach a temperature which will remain 
constant for the experiment. Should the temperature 
vary, it will be found in general better to repeat the 
observations than to correct for the variations, though, 
of course, the latter is possible. 

If the bridge wire is used with a shunt of relatively 
low resistance, the temperature of the shunt is of more 
importance than that of the bridge wire. In fact, if the 
bridge wire has n times the resistance of the shunt, a 
change of one degree in the temperature of the latter 
will produce n times as great a change in the value of p 
as would be produced by a change of one degree in the 
temperature of the former. 

In the second arrangement A and B were placed in 
parallel on one side, and C on the other. The difference 
between A and B in parallel and C was measured in 
terms of p. 



RESISTANCE. 69 

In the third arrangement B was removed, and the 
difference between A and C measured in terms of p. 
Let the bridge reading in these three arrangements be 
a, a' ; 5, h^ ; c, c\ Expressed in the form of equations, 
these three arrangements give the following relations : 

A—B=(a- a') p = mp, ... (1) 

4 7? 

^'"T^Tb^^*"*'^ '" = '*''' • ^^^ 

A-C=ic-e')p=pp; ... (3) 
adding (2) and (3), 

Eliminating B between (1) and (4), we obtain 

n + p± \/ Qti + p^(n + p — 7)1) 

To find which sign of the i is to be taken, substitute 
this value of p in (4). We obtain 
A 1 



V n -\- p 

From this it is evident that the plus sign should be 
taken, as otherwise B must be a minus quantity, which 
would be absurd. 

Consequently, 

w + i? + v(w + p)(^ + p — '?^0 

Example. 

A = 0.99795 [ 1 -f 0.00023 (19.3 — 15.4)] . 
a = 508.1. Coils A and B were at 19.3° C. 
a' = 497.25. The bridge wire was at 20° C. 
Whence, A — B^ 10.85^ = mp. 



70 ELECTRICAL MEASUREMENTS. 

b = 634.4. Temperatures as before. 
6' = 369.0. 

Whence, C- /-?-- = 265. 4p = np. 

A -\- B 

c =632.6. Temperatures as before. 
c' = 372.1. 
Whence, A — C= 260.5/) =_pp. 



Therefore, 
■95 [: 
525T9 + V'525.9 X 5lK05 



0.99795 [1 + 0.00023 (19.3— 15.4)] ^ ^,^,.,,, „^, ^, 
P = *- ^ , ^=0.00095459 at 20° C. 



43. Apparatus for exchanging the T-wo Coils to 
be compared. — Since the coils to be compared should 
be placed in water or oil baths, it is inconvenient to 
exchange their position from one side of the bridge to 
the other. A convenient and reliable de^T.ce for this 
purpose is a necessity. Fig. 29 shows one form which 
may be used in connection with a slide wire bridge by 
connecting with two binding-screws at one opening of 
the bridge. The connections are shown through the tT\^o 
commutators. If now both commutators are given a 
quarter turn, the circuits will be by the dotted lines, and 
it will be evident on tracing them that the two coils 
Si and So have exchanged sides on the bridge. 

An essential condition of such a commutating device 
is that the two sides shall be as perfectly symmetrical as 
possible, so that when the coils are exchanged unequal 
resistances are not exchanged along Avith them. An 
inspection of the diagram will show that the device is 
symmetrical. 

Connections are made by means of mercury cups. 
These should be of copper, with flat inside bottoms : and 
the copper rods composing the terminals of the coils 
compared, as well as the ends of the heavy copper links 



BESISTANCE. 



71 



of the commutators, should be well amalgamated, and 
they should be kept &mly pressed against the bottoms 
of the cups. Care should be taken to keep the amalga- 
mated ends of the rods clean. 






j 


o 


\ 




h 


o 


k 








1 1 




























— 



Fig. 29. 



The complete apparatus, shown in Fig. 30, contains 
the auxiliary coils S wound together non-inductively. 
They can be easily removed and others can be sub- 
stituted for them. The battery is connected to the 
binding-posts marked Ba. There are four mercury cups 
on either side for the purpose of placing two standard 
coils in parallel. Copper binding-posts are also provided 
for measurements not requiring the highest accuracy. 
The rods in each commutator are loosely mounted in a 



72 



ELECTRICAL MEASUBEMENTS. 



hard-rubber platform. They then adjust themselves to 
the bottom of the mercury cups, and good contact 
is secured. This apparatus may be used mth any form 
of bridge. 




Fig, 30. 

It is desirable to employ in the battery circuit another 
commutator, so as to reverse the circuit when the coils are 
exchanged, for the purpose of eliminating any possible 
thermal currents, or electromotive forces of thermal origin. 

Fig. 31 shows . the 
exchanging de^dce em- 
ployed by Mr. Glaze- 
brook in comparing the 
standard coils of the 
British Association. It 
is only necessary to 
move one coil up and 
the other down one 
step in order to have 
them exchange sides. 




BESISTANCE. 73 

44. The Calibration of the Bridge Wire — First 
Method. — The Carey Foster method itself may be 
applied to the calibration of the bridge wire. The cali- 
bration consists in laying off on the wire a series of 
exactly equal resistances. The process corrects not only 
for inequalities in the wire, but for errors of the scale as 
well. These inequalities and errors have thus far been 
neglected ; but they are always appreciable, though the 
error arising from neglecting them may be very small. 

It is CAddent that if the balance point for a given pair 
of coils aS'i and S-y can be shifted along the wire of the 
bridge by successive steps, and the readings ai and a2 
taken, the process will result in laying off equal resist- 
ances on the wire, each equal to jS\ — S.> . For this pur- 
pose take two resistance boxes of good adjustment for 
the auxiliary resistances M^ and II2 • Let the difference 
between the two coils Si and So be small enough to 
give convenient steps along the bridge wire. Adjust 
the auxiliar}^ resistances, which should be as large as 
the sensibility of the galvanometer will permit, till the 
balance point ai falls toward the zero end of the bridge 
wire. Since generally only a portion of the bridge wire 
near the centre will be used in the Carey Foster method, 
it is not necessary to calibrate it throughout its entire 
length. Find now by the exchange of the coils S^ and 
S2 the length of bridge wire having a resistance equal 
to their difference. Call this length li. Next shift 
resistance from M^ to Mo till with S\ and aS'^ in the first 
position the point of balance nearly coincides with the 
last point. It is not necessary to make these points 
agree exactly, though if they do the tabulation of the 
results is a little simpler. We shall assume for the 
present that the points do coincide, or that the distances 



74 ELECTRICAL MEASUREMENTS. 

/i, U^ etc., are end to end measurements. Now exchange 
Si and ^'2^ and by balancing again find ?., or a second 
length of the wire having a resistance equal to S^ — S.,- 
Reverse the coils, shift resistance from R^ to R, again 
till the beginning of the length of calibration 4 corre- 
sponds with the end of L. Then exchange coils and 
balance again to find ?,, . Continue the process till the 
required length of the bridge wire has been traversed. 
The balance fh'st obtained should be tested over again 
occasionally to be assured that S^ — S2 has not changed 
by reason of a change in temperature. These coils 
should be kept in a water bath to avoid changes of 
temperature as far as possible. It is equally important 
that the temperature of the bridge should remain con- 
stant. If any change in the length l^ occurs, the other 
values of I must be corrected in consequence. 

Now let the beginning of li on the scale read :c, and 
the end of the v}^ length read y. 

Then U + L+h^- . . . + 4 = ^ — 2;, and ^^I^ = Z, 
the mean length of calibration. 
Let l-l^ = ^^ 

4Z-(Zi+ . . . +h) = B,. 

nl-Q,-^ . . . +/,) = 8,^=0. 

S„ is necessarilv zero as / = ^^ — * — '- — '- — '—^' . These 

n 

quantities, 81, 8.,, 83, etc., are the corrections for the read- 
ings of the bridge wire. They are the amount which 



RESISTANCE. 15 

must be added algebraically to the readings to obtain the 
corrected readings. The correction for any length of 
the wire is the difference between the corrections at the 
ends of the length. The quantities S^, So, S3, may be 
either positive or negative. The plus and minus signs 
are used here in their algebraic sense. 







Example I. 










Bridge 




COBREOTED 


Correc- 


Resist. 


ANCE8. 


Readings. 


Lengths. 


Readings. 


tions. 






361.15 


• 


361.15 


0.0 


818 


1341 




12.0 










373.15 




373.113 


— 0.04 


816 


1277 




12.0 










385.15 




385.076 


— 0.07 


853 


1275 




12.05 










397.2 




397.039 


— 0.16 


848 


1210 




12.05 










409.25 




409.002 


— 0.25 


885 


1207 




12.15 










421.4 




420.965 


— 0.44 


885 


1151 




12.0 










433.4 




432.928 


— 0.47 


924 


1152 




11.9 










445.3 




444.891 


— 0.41 


923 


1101 




11.9 










457.2 




456.854 


— 0.35 


963 


1099 




11.8 










469.0 




468.817 


— 0.18 


962 


1051 




12.0 










481.0 




480.780 


— 0.22 


1001 


1047 




12.0 










493.0 




492.743 


— 0.26 


1000 


1000 




12.0 










505.0 




504.706 


— 0.29 


1045 


1000 




12.0 










517.0 




516.669 


— 0.33 


1046 


958 




11.9 







76 ELECTRICAL MEASUREMENTS. 







Bridge 




Corrected 


Correc- 


Resist. 


\.NCES. 


Readings. 


Lengths. 


Readings. 


tions. 






528.9 




528.633 


— 0.27 


1093 


958 




11.8 










540.7 




540.596 


-0.10 


1091 


915 




11.9 










552.6 




552.559 


-0.04 


1139 


914 




11.9 










564.5 




564.522 


-4- 0.02 


1140 


875 




11.9 










576.4 




576.485 


+ 0.08 


1192 


875 




12.0 










588.4 




588.448 


-f 0.05 


1193 


837 




11.9 










600.3 




600.411 


+ 0.11 


1246 


835 




12.05 










612.35 




612.374 


+ 0.02 


1243 


796 




11.95 










624.3 




624.337 


-f 0.04 


1281 


783 




12.0 










636.3 




636.3 


0.0 



Next let us suppose that the distances ^i, 4 5 h-, etc., 
overlap a little on the wire. As before let 



n 

and let l — l^ = hi 

21 - Q, + Z2) = h, 
U - (I, + ?2 + ^.0 = h 

etc. 

Also let r == tliE . 

n 

Then Si, S., S3, etc., are again the amounts which must 
be added to h, li + U-^ li-\- U^- h,^ etc., to make them 



r:esistance. 



77 



equal to I, 2/, 3/, etc. ; and supposing the overlap to be 
an insignificant part of each length, we may consider 8i, 
80, etc., to be the corrections from one end of the cali- 
brated portion of the wire up to the point considered. 
Strictly speaking, we should reduce these values Si, 89, 
S3, etc., in proportion to the amount of overlap. 







Example II. 




Bridge Readings. Lengths. 


Corrections 


2.80 . . . . • . 


0.0 


40.05 






37.75 




39.95 






. 


. -fO.lO 


77.95 






. 38. 




77.85 






. 


. — 0.04 


115.85 






38. 




115.75 








. —0.19 


153.65 






37.90 




153.55 








. —0.23 


191.45 






37.90 




191.15 . 








. — 0.28 


229.00 






37.85 




228.90 






. 


. - 0.28 


266.85 






37.95 




266.75 








. — 0.37 


304.75 






. 38.00 




304.45 






. 


. -0.52 


342.30 






. 37.85 




342.20 








. — 0.51 


380.00 






37.80 




379.9 






. 


. — 0.46 


417.85 






37.95 




417.8 








. - 0.55 


455.5 






37.70 




455.4 






, 


. — 0.40 


493.2 






37.80 




493.1 








. — 0.35 


530.85 






. 37.75 




530.75 






. 


. - 0.24 



78 



ELECTRICAL MEASUREMENTS. 


Bridge Readings. Lengths. Corrections 


568.25 . . . 37.50 


568.15 






. 4-0.11 


606. 






37.85 


605.9 






. +0.12 


643.85 






37.95 


643.75 






. +0.02 


681.45 






37.70 


681.30 






. +0.17 


719.30 






38. 


719.50 






. +0.02 


756.85 






37.35 


756.8 






. +0.53 


794.9 






38.10 


794.8 






. . +0.28 


832.75 






37.95 


832.65 






. +0.18 


870.40 






37.75 


870.30 






+ 0.29 


908.45 






38.15 


908.35 






. —0.01 


945. 






37.65 


944.95 






. +0.20 


983. 






38.05 ... 0.0 


Mean 


. 37.854 



The successive points at which the correction should be applied 
are l\ 2^' 3Z' etc. 



45. Calibration of Bridge Wire — Second Method.^ 

— Make as many approximately equal resistances as there 
are steps in the desired calibration. Let this number be 
n. Fig. 32 shows ten such resistances. Let them con- 
nect the mercury cups 1, 2, 3, etc. To insure good con- 
tact each small resistance should be soldered to a short 



Cavl Barns, Bulletin U.S. Geological 



, Xo. 14. 



RESISTANCE. 



79 



heavy rod of copper. If L is the length of AC to be 
calibrated, and /' the interval of calibration 



= 1, 



Find a point 3Ii on AB having the same potential 
as iVi , and JjT. the same as iV^, . This is done by means 
of the sensitive galvanometer G. 

Then exchange wires Nos. L»and II. Find points on 
AC having the same potential as JSF^^ iV^3, respectively. 
Call these points M2, M.. The resistance of I. should 




Fig. 32. 

be such that the reading for M'^., iff '3, etc., shall be a little 
smaller than for ilf^, Jfy, etc. That is, the calibration 
distances set off should overlap a little. 

Then exchange I. and III. and perform the same oper- 
ations as before. Continue the process till the conductor 
I. has been carried along the entire series and finally 
takes the place of the last one. The result is to lay off 
along the bridge wire distances such that the P.D. 
between their ends is the same as between the ends of 
conductor I. If the current remains absolutely constant, 
all these potential differences are equal to each other, 
and therefore the resistances of the successive lengths 
laid off are also equal. They will equal one another if 
the current does not remain constant, provided the rela- 



80 ELECTBtCAL MEASUREMENTS. 

tive resistance of conductor I. to this part of the divided 
circuit remain the same ; for any decrease in the current 
will cause a decrease in the P.D. between A and C, and 
this P.D. is the same in going from A to Chj either 
path. So long therefore as conductor I. bears the same 
ratio to the entire resistance of the path of which it 
forms a part, the resistance between the points Mi, iHf^, 
M2 ^ M-i^ etc., will be the same. The eifect is then to lay 
off a series of equal resistance lengths on A (7, and these 
lengths overlap somewhat. 
Then we have as before 

n ' 

and the results are treated in the same way as by the 
other method. The corrections will be 

At V, + 3i 

etc., etc. 

46. Measurement of the Temperature Coefficient. 

— The Carey Foster method of comparing resistances is 
especially adapted to the measurement of the variation 
of the resistance of a conductor with temperature. The 
process consists in comparing the resistances of two coils, 
one of which is maintained at an unvarying tempera- 
ture, while that of the other is changed. The resistance 
which is maintained constant ma}" be a standard coil, and 
the other is made of the wire or conductor to be investi- 
oiited. Both of them must be immersed m a bath; the 
one in order that the temperature may remain invariable, 
and the other that its temperature may be varied and 



RESISTANCE. 



81 



accurately measured. The equation expressing the re- 
sistance of a conductor at any temperature is, to a first 
approximation, 

i^, ^i^oCl + aO- 
If now the resistances of tlie conductor under test at 
temperatures t^ and t, are Bi and B., -> then 

B, = Bo (1 + atO 

and Bo = Bo (1 + at.). 

Subtracting, Bi — Bo = Bo a (^i — t.) 

Bi Bo 1 
a = — — = . . 

Bo t] — to 



and 



Bo does not need to be known with great accuracy, for 
a and Bi — Bo are very small ; and when the numerator 
of a fraction is relatively small, a small change in the 
denominator produces only an inappreciable change in 
the value of the fraction. A first or approximate value 
of a may be found, and this value may be used to find 
the value of Bo with sufficient accuracy. A second 
approximation will then give a nearer value of a. 

Example. 

Temperature Coefficient of a Coil of Platinoid Wire. 



Standard 
Coil {S). 


1 
Bridge Wire. 


JT—S. 


Tested Coil (X). 


is 


Resist- 


%s 


READINGS. 




cT 


Resist- 


2S 

a; C- 


ii 




ance 


p.^ 






PXIO* 




ance 


^t 












B 


(ohms). 


a 2 

18.1 


Oi 


"2 


a^-a. 




8^ 

Q. 


(ohms). 


16.7 




15.2 


9.8666 


■557.0 


442.5 


114.5 


9.538 


0.1090 


9.9756 




15.2 


9.8666 


18.3 


563.4 


435.6 


127.8 


9.538 


0.1218 


9.9884 


23.1 


0.00200 


15.2 


9.8666 


18.4 


570.1 


429.0 


141.1 


9,539 


0.1345 


10.0011 


29.1 


0.00216 


15.2 


9.8666 


18.4 


576.3 


423.3 


153.0 


9.539 


0.1459 


10.0125 


34.9 


0.00196 


15,2 


9.8666 


18.5 


581.2 


418.0 


163.2 


9.540 


0.1557 


10.0223 


40.3 


O.OOISI 


15.4 


9.8672 


18.6 


586.5 


412.6 


173.9 


9.540 


0.1659 


10.0331 


45.2 


0.00220 


15.4 


9.8672 


18.7 


589.5 


409.4 


180.1 


9.540 


0.1718 


10.0390 


48.1 


00203 


15.7 


9.8681 


18.8 


595.0 


403.5 


191.5 


9.540 


0.1827 


10.0508 


54.0 


0.00200 



82 



ELECTRICAL MEASUREMENTS, 



Total increase in X= 10.0508 — 9.9756 = 0.0752 ohm. 
0.0752 



Increase per degree 



54 — 16.7 



0.002016. 



Resistance of X at 0° C. == 9.9756 
ohms. 

^_ 0.002016 
9.9409 



16.7 X 0.002016 = 9.9409 



Therefore, 



0.000203. 




Fig. 33. 



47. The Conductivity 
Bridge . — The C a r e y 
Foster method furnishes 
an elegant means of 
measuring very small re- 
sistances, such as the 
resistance of metal bars, 
rods, and the like. For 
this piupose a special 
piece of apparatus is re- 
quired. Its principle is 
precisely the same as that 
employed in finding p 
from a known difference 
of resistance. The rod 
or bar to be measured 
takes the place of the 
bridge wire, and its con- 
ductivity is found by lay- 
ing off a length of the 
rod equal in resistance to 
a known resistance repre- 
sented by accurately ad- 
justed coils in parallel. 
The bar to be measured is 
held securely by clamps 
D (Fig. 33). It is parallel 



RESISTANCE. 



83 



to a scale U, which is read by a vernier to l-20th mm. 
The sliding contact may be clamped by the screw M to 
the rod which carries it, and a slow motion may then be 
given to it by the nut J working against the spring Y. 
The commutator K commutes both the known resist- 
ances and the battery, either simultaneously or sepa- 
rately. The adjusted coils are inserted in parallel by 
means of heav}^ copper links dipping into suitable mer- 
cury cups in large masses of copper. The battery and 
galvanometer are connected by means of binding-posts 
at the back of the instrument. The method of operation 
is precisely the same as in the Carey Foster method. A 
known difference of resistance is laid off on the bar to 
be tested, and the length of the bar between the two 
contacts is measured by means of the scale and vernier. 
The measurement is independent of the contacts on the 
bar. 

48. Insulation Resistance by Known Potential 
Differences. 1 — This method of measuring a high resist- 



ance consists in comparing the current sent by a given 
P.D. through it with that sent through a known resist- 
ance by a fraction of this same P.D. A potential 



Ayrton's Practical Electricity, p. 278. 



84 



ELECTRICAL ME A 8 UREMENTS. 



difference may be subdivided into known fractions by 
causing a steady current to flow through a very high 
resistance with known subdivisions. Then the P.D. 
between any two points ST (Fig. 34) bears to the P.D. 
between the points ML at the extremities of the 
high resistance the same ratio that the resistance of 
the part ST bears to the whole resistance ML. 

Let the entire P.D. between L and M be employed to 




w 



Fig. 35. 

send a current /^ through the unknown high resistance x 
and the galvanometer Gr (Fig. 35). The galvanometer 
must be one of the highest sensibility. Next let the 
P.D. between L and T (Fig. 36) be employed to send a 
current through the known resistance r, and the galva- 
nometer shunted with resistance s; r must be large 
with respect to q. Let the current through the galva- 
nometer be Ii. 



Then, 



L_ q x + g 



s + .g 



RESISTANCE. 



85 



Whence, x+g = ^ .P(r + ^ii^, 



M/WWWVt 




Fig. 36. 



If 



S(^ 



may be neglected in comparison with r, and g 



s +9 
in comparison with a;, then 



07= -^ . ^ .r 
Ii q s 

or, if Ji and I2 are proportional to the deflections of the 
galvanometer in the two cases, 

di q s 

Example. 

r = 250,000 ohms ; p = 10,200 ohms ; rfi = 48.2 ; 

s = I ; ^ = 200 ohms ; d2 = 38.0. 



10,200 X .S8.0 X 250,000 X 10 
200 X 48.2 



100.5 X 106 ohms 
= 100.5 megohms. 



86 



ELECTRICAL ME A S UEEMENTS. 



49. Insulation Resistance by Direct Deflection. — 
When the constancy of the battery cannot be relied on, 
it may be found advantageous to proceed as follows : 
First find the figure of merit of the galvanometer (Art. 
29), i.e.^ the current which will produce a deflection 
of one division of the scale. The galvanometer then 
becomes an ammeter, and may be used in connection 




Fig. 37. 



with a voltmeter V.M. (Fig. 37) to measure the un- 
known resistance x. If the y'-q, xJ-d-^ or yoVo shunt is used 
with the galvanometer, its figure of merit is correspond- 
ingly increased. Let F equal the figure of merit, d the 
deflection with x^ as in the figure, Fthe number of volts 
shown by the voltmeter, and g the resistance of the 
galvanometer. Then the current is Fd^ and by Ohm's 
law 



9 + x = ^,or 



V 



RESISTANCE. 



87 



Example. 

Test of a Piece of Common Line Wire. 
Diameter over insulation 8.2 mm. 
Diameter of bare wire 4.13 mm. 
Length under water 90 ft. 

r = 250,000 ohms. E.M.F. of Clark .cell 1.431 volts. 



-1 
99 



d\ = 143.4 mm. 

Figure of merit (with shunt) 



1.434 



= 0.000,000,04 



143.4 X 250,000 
ampere per mm. = 0.04 micro-ampere per mm. 

Figure of merit (without shunt) = 0.0004 micro-ampere per 
mm. 



Time after 


Volts 


Deflections 


Current 


Insulation 
resistance 


Resistance 


immersion. 


shown by 


in 


in 


in megohms 


h. ra. 


voltmeter. 


mUlimetres. 


micro-amperes. 


megohms. 


per mile. 


: 00 


50.2 


176 


0.0704 


713 


12.2 


05 


50.2 


165 


0.066 


761 


13.0 


10 


50.2 


160 


0.064 


784 


13.4 


15 


50.2 


160 


0.064 


784 


13.4 


27 


50.2 


177 


0.0708 


709 


12.0 


30 


.50.2 


184 


0.0736 


682 


11.6 


1 : 00 


50.2 


232 


0.0928 


541 


9.2 


2 : 35 


50.2 


670 


0.268 


187 


3.2 


5 : 00 


50.5 


1925 


0.77 


65 


1.12 


27 : 00 


66.5 


38000 


15.2 


4.37 


0.075 



In the column of "Deflections in millimetres," the larger num- 
bers are the products of the deflections and the multiplying power 
of the shunt. 



60. Insulation Resistance by Leakage.' — The 

method consists in charging the cable as a condenser, 
letting it leak for a few observed seconds, and then 
charging to the full potential again by connecting 
through the galvanometer. 

^Electrical Engineer, May 20, 1891, p. 565. 



88 



ELECTRICAL ME A S UBEMEN TS. 



First. To find the constant of the ballistic galva- 
nometer G- (Art. 97). This may be done in two ways. 
The first consists in charging a condenser of knoAvn 
capacity by a known E.M.F., and then discharguig 
through the galvanometer. Let the apparatus be set up 
as shown in Fig. 38, in which ^ is a charge and dis- 
charge key, C is 
the condenser, 
and Gr the galva- 
nometer. The 
battery B may 
be a standard 
cell, the E.M.F. 
of which is 
known. Then if 
Q is the quantity 
of electricity dis- 
charged through 
the galvanom- 
eter, O the capacity of the condenser, and ^i the E.M.F. 
of the cell. 




Fig. 38. 



If the deflection is di , 



and 






k = 



^1 



The other method^ involves the exact measurement 
of a current. A long magnetizing coil is uniformly 
wound on a wooden cylinder or other non-metallic core, 
the diameter of which is accurately known. Over this 



^ Ewiag's Magnetic Induction in Iron and other Metals^ p. 62. 



BESISTANCE. 89 

primary, at the middle of its length, a short secondary 
coil is AYOund and put in circuit with a ballistic galva- 
nometer. 

Let A be the mean area of cross-section of the primary 
coil, and let ^i be the number of turns in it per cm. 
length. Then if a current of /amperes be made to pass 
through the coil, the magnetic flux or induction within 

it near the middle is per square centimetre,^ and 

the total number of lines of induction within the coil is 
4:7rInA 



10 

If iV is the number of turns in the secondary and r 
the resistance in the circuit of the galvanometer, then 
the quantity of electricity in coulombs passing dur- 
ing the flow of the transient current in the secondary, 
when the primary circuit is made or broken, is 

r X 10"' 

The first method requires a knowledge of capacity and 
E.M.F. ; the second requires a knowledge of current and 
resistance in addition to the dimensions of the coil. 

Second. The operation with the cable as a condenser. 
The apparatus must be set up as indicated in Fig. 37. 
The coil is immersed in water contained in a tank T, 
lined with sheet copper. P is a short-circuiting key. The 
entire circuit should be as well insulated as possible ; but 
in any case particular care should be taken to insulate the 

1 Stewart and Gee's Practical Physics, Part II., p. 328. 



90 ELECTRICAL MEASUREMENTS. 

ends of the coil. The end which is not used should be 
sealed, and there should be enough of the coil out of 
water at both ends to avoid leakage along the surface. 
If an additional wire is used to connect the coil to the 
key, great care must be taken to insulate it. It may be 
suspended by a silk thread. The insulation of the key 
when open should also be very good. A charge and 
discharge key is satisfactory for this purpose. 

Then with the switch at P closed, charge the coil as a 
condenser by pressing the key K. Since a part of the 
charge is absorbed, constant results will not be obtained 
unless the key be kept closed for a long time, several 
hours at least. If the usual rule of one minute be 
adopted, the insulation resistance will appear to be lower 
than it really is. However, on the first test of an insu- 
lated wire it is not advisable to attempt to obtain con- 
stant results from the start, as poor insulation may 
completely fail before such a condition is reached. There- 
fore, if a first test is being made, charge the coil for a 
short time with P closed ; next open the circuit for an 
observed number of seconds, and meanwhile open P. 
Then again close K^ thus causing the quantity of elec- 
tricity §, required to replace the part of the charge 
AAdiich is lost by leakage or is absorbed, to pass in 
through the galvanometer. Let d.2 be the deflection 
produced by §, and JE.2 the E.M.F. of the charging bat- 
ter}^ If we make no allowance for the part absorbed, 
the integral of the leakage current /for the time t must 
equal Q, 

Then Q = /ldt=/^dt, 

in which R is the insulation resistance sought. If 



EESISTANCE. 91 

dining the time of leakage the difference of potential 
has fallen a negligible amount only, then 

K 

^=¥-^- 

Substituting the value of k from the first method, and 

Ty_E2 dy t_ 

~ E,' d,' C' 

If we use the value of k obtained by the second method, 

7-> J^2 di rt w Q9 

" 12366 * J, ' In AN ' 

If C in the first formula above is in microfarads, M 
will be expressed in megohms. 

In the second if I is in amperes, JR. will be in ohms. 

Example. 

Test of a Piece of Orimshaw Wire. 
Diameter over insulation 5.6 mm. 
Diameter of bare wire 2 mm. 
Length under water 200 ft. 
C =0.1 microfarad, di= 129 mm. 

Ex = 1.44 volts, h = 0.00112 micro-coulomb per mm. 

E-i = 57 volts throughout the test. 



92 



ELECTRICAL ME A S UREMENTS. 



Time after 

immersion. 

h. m. 8. 


Intervals in 
seconds. 


Deflections 
in mm. 


Deflections 
Intervals. 


Insulation 
resistance in 
megohms. 


: 30 










1 : 00 


30 


43 


1.433 


35780 


1 : 30 


30 


25 


0.833 


61540 


2 : 30 


60 


34 


0.567 


90500 


3 : 30 


60 


24 


0.400 


128100 


4 : 30 


60 


18 


0.300 


170800 


5 : 30 


60 


15 


0.250 


205100 


6 : 30 


60 


14 


0.233 


219800 


7 : 30 


60 


11 


0.183 


279700 


13 ': 30 


'.'.'.'. 


.... 


.... 




15 : 30 


120 


15 


0.125 


410200 


17 : 30 


120 


14 


0.117 


439500 


19 : 30 


120 


12 


0.100 


512400 


58 : 00 


.... 


.... 


.... 




1 : 00 : 00 


120 


8 


0.067 


769200 



The charging of the cable was begun thirty seconds after 
immersion. 

This example gives a good illustration of the absorp- 
tion of the charge by an insulated wire. This absorption 
will sometimes continue for hours ; and if the insulation 
is really waterproof, the highest value — which is the 
real value — will be obtained only by electrifying the 
wire until the absorption ceases. 



51. Second Method of Insulation Resistance by 
Leakage.* — This method is particularly applicable to a 
resistance having capacity, such as a cable immersed in 
water. Let this capacity be C microfarads. Let V be 
the P.D. between the two surfaces at the instant when 
the charge is Q. Then 



Q= CTand 



dQ_^dV 



dt 



C 



dt 



Gray's Absolute Measurements in Electricity and Magnetism, p. 253. 



BESISTANCE. 



93 



But ^ z= — = I, where E is the unknown resistance 

dt R 

through which the charge leaks. Therefore, 

dt 



^dV^ V ^ dV^ 



0. 



Integrating, 
If theP.D.. 



log^ Y+ -^^ = constant. 

Vo when ^ = 0, then 
logg Vi, — constant^ 



and 



or 



loge F[) — logg V= 



OR' 



E = 



C 






lORe-f 



To determine the ratio of FJ, and FJ the coil or cable 
is charged as a condenser, and then immediately dis- 
charged through 
a ballistic galva- 
nometer, and the 
deflection is 
noted (Fig. 39). 
The coil is again 
charged to the 
same potential 
as before, and is 
then insulated 
and allowed to 
leak for an ob- 
served number 




^ This equation may be put into the form F= Voe~iic, and this last 
expresses the law according to which the potential of a condenser varies with 
the time. 



94 ELECTRICAL MEASUREMENTS. 

of seconds ; and finally it is discharged through the gal- 
vanometer. The deflections, if moderately small, are 
taken proportional to the P.D.'s of the coil at the times 
of discharge. If the capacity is expressed in microfarads, 
and common logaritlnns are used in the reduction, then 

t 1 



R^W 



^ logio^^x 2.303 



where R is expressed in ohms. But if it is desired to 

express R in megohms, then the multiplier 10*^ is omitted. 

The chief difficulty with this method arises from the 

absorption of the charge by the dielectric. The second 

deflection may in consequence be larger than the first. 

This difficulty may be avoided in part by first charging 

the cable and allowing it to leak for say twenty seconds, 

and then discharging through the galvanometer. Then 

charge again and allow the leakage to extend over a 

longer period — say forty seconds — and then discharge 

again. The ratio of the deflections may then be taken 

as the ratio of the potential differences Vo and J\ the 

time t being the difference in seconds of the two periods 

of leakage. 

Example. 

Obser^vations : A coil of 1000 ft. of insulated wire was charged 
with one cell, and the discharge through the galvanometer gave 
a deflection of 123 mm. 

The coil was again charged, and after leaking 120 seconds the 
deflection was 115.8 mm. (as a mean of five observations). 

The capacity of the coil was 0.082 microfarads (Art. 97). 
Calculation : 

^^120 1 120 1 



0-082 1 _m 0.082 iog,,i2Lx 2.303 
"= 115.8 ""^15.8 

or 7? = 2.4251 X 10^ megohms. 

Therefore the resistance per mile is 24251 ^ 5.28 = 4593 meo-- 
ohms. 



RESISTANCE. 



95 



52. To measure a Resistance by the Fall of Poten- 
tial. — Let AB be the resistance to be measured (Fig. 
40), and let an ammeter Am be placed m series with 




Fig. 40. 

it. Let Vm be a voltmeter of high resistance to measure 
the P.D. between A and B. Read simultaneously the 
two instruments. Let J be the current and Fthe poten- 
tial difference between A and B. Then by Ohm's law 

Example I. 

Required the Resistance of the Secondary of a 12.6 Kilowatt 
Transformer. 
Apimratus : A milU-ammeter and a milli-voltmeter. The 
resistance of the milli-voltmeter was relatively high compared 
with the resistance to be measured. The scale read both ways 
from the centre. Hence to eliminate errors of the scale and zero, 
the milli-voltmeter was read first on one side and then on the 
other. Also the current was reversed through the resistance. 
Amperes. Volts. 

1.235 <; 0060 Direct. 

1.249 } .0061 Reversed. 

1.245 } .0060 Reversed. 

1.255 <; 0062 Direct. 

1.250 \ 0060 Direct. 

Means, 1.2468 .00606 

.-. i? = -^2606 _ 00486 ohm. 
1.2468 



96 



ELECTBiCAL MEASUREMENTS. 



Example II. 

Measurement of the Resistance of an Edison Lami^. 
The observations of volts and amperes were made with the 
lamp at the given candle-power ; the resistance of the lamp was 
then calculated for each set. 







Obsekved. 




Calculated 








Resistances. 


Candle Power. 


Volts. 


Amperes. 




.79 


34.6 


.710 


48.7 


1.30 


37.1 


.770 


48.2 


1.84 


39.4 


.824 


47.8 


3.71 


42.5 


.920 


46.2 


7.26 


47.0 


1.028 


45.7 


13.85 


50.3 


1.140 


44.1 


20.34 


54.0 


1.240 


43.5 


31.13 


57.3 


1.350 


42.4 


35.20 


58.7 


1.380 


42.5 



53. To measure the Internal Resistance of a Bat- 
tery — First Method. — The following method of meas- 
uring the internal resistance of a battery is specially 
applicable when this resistance is very small, as in the 
case of a secondaiy cell, or a series of such cells. It 
requires a suitable voltmeter and ammeter with a 
resistance to give the current a convenient value. 

Let B be the battery (Fig. 41), Vm the voltmeter. Am 
the ammeter, R the resistance in the circuit, which need 
not be known, and let r be the internal resistance to be 
measured. Fii-st measui-e the P.D. between the termi- 
nals of the battery with the key jfiTopen, and let it be 
represented by E, Then close the key, and read simul- 
taneously and quickly both Am and Vrn^ and let the 
current and P.D. be Zand E'. Then 



E=E+Ir, 



HESISTANCK 



9T 



in which Ir is the loss of potential within the cell due to 
current /passing over the resistance r, and U is the fall 
of potential over the entire circuit. 

W hence r = 

If the battery consists of several cells, r is the sum of 
the internal resistances of the series. 




ittiiiill 




/Vv 



Fig. 41. 

In the case of a storage battery this method may be 
slightly modified by measuring the charging current and 
the P.D. between the terminals of the battery simulta- 
neously ; and then, after opening the circuit, measuring 
the P.D. or E.M.F. again. Then if U^ is the P.D. 
during charging, JS the E.M.F. of the battery on open 
circuit, I the charging current, and r the internal resist- 
ance of the series of cells, 

since the difference between the two voltages is the 
E.M.F. required to maintain the current I through the 
resistance of the batter}^ 



98 



ELECTRICAL MEASUREMENTS. 



Example. 

It was desired to find the internal resistance of a storage battery 
of 36 cells. The battery was joined up in series with an ammeter 
and sufficient resistance to give {a) 5 amperes and (b) 10 
amperes. The voltage of the battery was measured while giving 
these currents, and immediately afterwards on open circuit (ex- 
cept for the voltmeter of 19,560 ohms resistance). 

Internal resistance. 



0.10 
0.09 



(a). 


^'Yiuperes. 

5 


71.5 







72. 


(6). 


10 


70.9 







71.8 
Mean 


istance 


of each cell. 


0.0026 ohm. 



0.095 



54. Battery Resistance — Second Method- — Form 
a circuit with the battery and a high resistance of 10,000 
ohms or more (Fig. 42). Let a derived circuit be taken 

from two points 
on this high re- 
sistance with 
o n 1 y a small 
part of the whole 
resistance be- 
tween them ; or 
a small a d d i - 
tional resistance 
i?i may be added 
to the high re- 
sistance, and the 
derived or shunt circuit may be joined up round this so 
as to include a d'Arsonval galvanometer Gr, as shown in 
the figure. If the galvanometer is a sensitive one, the 
resistance J?i will be so small that no shunt to render 



©• 






]- 


-• 10,000 ohms •"" 


J — I R, 






^1 




|l i 




L_^ R 


*■ K 






Fig. 42. 



EESISTANCE. 99 

the galvanometer ''dead beat" will be required. A 
circuit is also formed so as to close the battery through 
a small resistance H of from one or two to five ohms. 

Proceed as follows : Let di be the deflection of the 
galvanometer when the circuit is closed through the 
high resistance, the key K being left open ; and let do be 
the deflection when key K is closed. The two deflections 
are proportional to the currents through the galvanometer, 
and therefore to the P.D.'s at the terminals of i^i, 
with K open and closed respectively. Since Mi bears 
a constant ratio to the entire resistance in circuit, the 
deflections di and d-y are proportional to the P.D.'s at 
the battery terminals in the two cases. 

Hence d,: d,:: E : U' :: B,+ r : E. . . . (1) 

When the key K is open the P.D. at the battery ter- 
minals, measured by t/i, is the entire E.M.F. of the cell 
if its internal resistance is negligible in comparison 
with the high resistance in circuit; and when IC is 
closed the P.D. measured by do is the fall of potential 
over the external resistance H. Now if the E.M.F. of 
the cell does not change immediately on closing K^ then 
the fall of potential over the entire resistance H + r 
is the E.M.F. of the cell. We may, therefore, put the 
two deflections proportional to the two resistances. 

From (1) by subtraction, 

di — dj : do :: r : R. 

Whence, r=B,^-l^lAl. 

do 

It is necessary to use a "dead beat" galvanometer, or 
one which swings back to zero or takes a deflection 
corresponding to the current through it Avithout swing- 



100 



ELECTRICAL MEASUREMENTS. 



ing back and forth, in order that the reading for d2 may- 
be taken quickly after closing IC^ and before polarization 
has changed the value of the E.M.F. of the cell. The 
d'Arsonval galvanometer is therefore recommended for 
this 23urpose. 

Example. 





R 


di 


d2 


r 


Daniell cell .... 


. 5 


64 


35 


4.14 


" . . . . 


. 10 


64 


45 


4.22 


Gassner's dry battery . 


. 5 


75 


22 


12.04 


" 


. 10 


74 


33 


12.42 




55. The Condenser Method of measuring- Battery 
Resistance. — Let B be the battery to be experimented 
upon (Fig. 43), C a condenser, and K a charge and dis- 
charge key, discharging 
on the upper contact. 
When K is depressed 
the battery charges the 
condenser ; when K is 
released and makes con- 
; tact on the upper point, 
the battery is discon- 
nected and the condenser 
is discharged through 
: the galvanometer Gr. 
This must be ballistic 
or slow-swinging, so that 
the first swing may be 
easily read ; and it must have but little damping. 

The operation consists in charging and discharging, 
first with the second key Ki open, and then with it closed, 
and noting the deflections di and d2 . 

The deflections are taken as proportional to the quan- 



Fig. 43. 



BESISTANCE. 101 

titles of electricity discharged If they are not too large, 
and these quantities are proportional to the two P.D.'s. 
Hence the deflections are proportional to the P.D.'s and 

E 









d. 


: d. 


: : E + r 


as 


i" before 
Also 


(Art. 


54). 


r = 


j^d,-d, 

di 



The key K can be operated so quickly that Ki need 

not be kept closed long enough to permit appreciable 
polarization. 

Example. 

R d\ ck r 

Gassner's dry battery ... 5 130 66 4.85 

Crowdus dry battery .... 1000 83 47 766.6 

In the case of the Crowdus battery 5 ohms were tried at first, 
but no appreciable deflection was obtained for d^ , sliowing that 
the internal resistance was extremely large in comparison with 5 
ohms. The cell was an old one nearly exhausted. 

36. Value of R for Least Error. — To determine 
the conditions of highest accuracy it is necessary to con- 
sider the effect of an error in observing both d^ and 6?2- 

Emplopng the general principle of Art. 36, find first 
the partial derivative of r with respect to t/s • It will 
have the minus sign, because r Increases as 6?2' decreases. 
From the equation 

r=E'^::^\ 

di 

we have F=-f%=Bf^, 

oa2 di 

but R = r '^ 



h-d. 



102 ELECTRICAL MEASUREMENTS, 

Hence, F = rf ^L__ . 

do (c?i — c^s) 

FinaUy, ^=./ ^' 



doXdi-fh} 

This is the relative error in r due to an error / in 
observing do. It is a minimum when the denominator 
is a maximum, since di is now considered constant. 
But the denominator consists of tAvo factors Avhose sum 
is a constant, or 

do + (t-?i — c?.) = ^1- 

Now, when the sum of two factors is a constant their 
product is a maximum when they are equal to each 
other, or in this case, when do = di — do , or when d. = Jc?i . 
This means that B should be equal to r. 

To estimate the influence of an error/ in tZi, find the 
derivatiA'C of r mth respect to di. 



bdi " d^ do (d-^ — d.j) 

Since this expression has the smallest value when 
c?2 = 0, or when the cell is short-circuited, the condition 
is inapplicable. 

In case the errors in d^ and d2 are equal and of the 
ojjjjosite sign^ then adding the corresponding values of 
the resulting errors, 

I[^ di+d. 
rf~ d,(d,-d.:;)' 

To fuid when this is a minimum, consider di constant 
and differentiate the fraction with respect to do thus : 

b /F\_d2(d,- dO - (d, + d,) (d, - 2d.;) ^ ^^ 



^. \rf) 



^^2 \rf I dl {d^ — d^y 



RESISTANCE. 103 



Hence (d,-d;f=m. 

or di — d> = (io\/2 . 

Therefore, d^ = d, (1 + \/5) -= 2AU2d.2 , 

or <^o — ^ 



2.4142 



The resistance B shonlcl then be about f r. 
Finally, if the equal errors in di and c?2 are of the 
same sign, then 

F _ d,-d, 1 

rf d.> (^di — (^2) d.2 

This expression is a minimum when d^ is greatest; 
that is, when c?, = <^i 5 or when the external resistance is 
infinite. This is clearly an impossible condition. 

In this particular problem an error in d-^ is much less 
likely to occur than in d^ . A series of readings can be 
taken with the battery circuit open, and the mean will 
be c?i . But c?2 is dependent to a considerable extent on 
skill in manipulation, and is affected by polarization ; 
hence an error in it is much more likely to occur than 
in di . It appears better to consider d., only as the 
variable. The result is that M should equal r for high- 
est accuracy. 

The problem has been solved usually on the assump- 
tion that if the errors in di and d^ are of opposite sign, 
the resulting error F will be a maximum ; and the con- 
dition is then found for the relation between B and r 
which gives the smallest value of F. The result is 

But here a special assumption is made and a general 



104 ELECTRICAL MEASUREMENTS. 

conclusion is drawn. There is no good reason for the 
assumption that the errors in di and cl.y will be equal, 
and especially of opposite sign. 

A preliminary measurement of the resistance r can 
first be made, and then a second one, with R nearly the 
same as the preliminary value obtained for r. If r is 
quite small this should not be done, since a small 
external resistance will permit rapid polarization, and 
the error thus introduced may be greater than the one 
we seek to avoid. 

In general, therefore, the principle can be applied only 
to batteries of high internal resistance, or to those which 
do not polarize rapidly. 

57. Variation of Internal Resistance with Current. 
— The internal resistance of a voltaic cell, even at a 
constant temperature, has not a fixed and definite value, 
but depends upon the current flowing through it. The 
preceding methods of measuring this internal resistance 
enable one to determine what is the available potential 
difference at the battery terminals with a given resist- 
ance in the external circuit, or with a given cuiTent flow- 
ing. The resistance measured is a quantity satisfying 
the equation 

T^E-E' E E' r 

where r is the internal resistance corresponding to a 
current I. 

To determine the dependence of r upon J, the con- 
denser method may be employed, using different external 
resistances in succession. The examples following illus- 
trate the great variation in r which is sometimes found : 



BESISTANCE. 



105 



Example I. 

Gassner's Dry Battery. 
E = 1.213 volts. 



^1 


dl 


R 


r 
21.5 


/ 


i71 


258 


400 


.0028 




249 


200 


17.7 


.0056 




238 


100 


13.86 


.0106 




227 


50 


9.69 


.0203 


271 


223 


40 


8.6 


.0249 




218 


30 


7.3 


.0324 




212 


20 


5.56 


.0473 




204 


15 


4.93 


.0607 


■ 


194 


10 


3.96 


.0868 


271 


172 


5 


2.87 


.1538 




164 


4 


2.59 


.1838 


270 


153 


3 


2.29 


.2289 



Example II. 

Daniell Cell. 
E=l.l volts. 



dl 


d.2 


B 


r 


/ 


246.5 


216.4 


40 


5.56 


.024 




208.7 


30 


5.43 


.031 




194.2 


20 


5.39 


.043 




181.8 


15 


5.34 


.054 


246.5 


161.8 


10 


5.24 


.072 




148.4 


8 


5.29 


.083 




132. 


6 


5.21 


.098 




121.6 


5 


5.14 


.109 


246.5 


108. 


4 


5.13 


.121 




91.8 


3 


5.06 


.137 




70.5 


2 


4.99 


.157 


246.5 


42.5 


1 


4.80 


.190 



106 



ELECTRICAL MEASUREMENTS. 



These results are plotted in Fig. 44, with internal resistances 
as ordinates and currents as abscissas. The Gassner cell shows 
a much larger decrease in the internal resistance than the Daniell 
cell for the same range of current. The scale of internal resist- 
ances for the Daniell is twice as large as for the Gassner. 





\ 


























\ 
























16 


\ 
























\ 
























12 


A 
























8 \ 




_^ 




2 a 

















8 




V 
























\ 


^ 
















i 






^^""^^ 










1 


















6- 


















Ajnj 


jeres 




1 







.02 



.06 .08 



.10 .12 .14 
Fig. 44. 



.16 -.18 



.20 



.22 



.24 



58. Auxiliary Apparatus for the Condenser 
Method. — In applying the condenser method to the 
measurement of internal resistance, or to the determina- 
tion of polarization in an electrolyte, it is essential for 
quantitative comparison that some mechanical means be 
adopted to control the time during which the circuit is 
kept closed. It is perhaps equally important that the 
condenser should be discharged as soon as possible after 
charging, and before it has lost appreciably by leakage. 
The pendulum apparatus of Fig. 45 meets the require- 
ments admirably. For the principle employed the authors 
are indebted to Dr. Mihie ^Murray, of Edinburgh. 



BESISTANCE. 



107 



A rectangular frame carries at the bottom a heavy 
pendulum bob adjustable in height. The time of vibra- 
tion of this pendulum is about one second. The bob is 
held in place by a detent in the position shown. When 
it is released it swings between two parallel circular arcs, 
concentric with the axis 
of suspension. The dis- 
tance apart of these arcs 
is a little less than the 
length of the lower cross- 
bar carrying the heavy 
bob. They support four 
keys, which can be 
clamped at any desired 
points. The keys have 
an upper and a lower 
contact like a simple 
discharge key. When 
the key lever is erect, the 
key makes contact on 
the lower point; and 
when the lever is thrown 
over by the crossbar of 
the pendulum' as it 
swings forward, the key 
makes contact on the upper point. These keys can be 
set in any relation to one another which may be desired, 
and their operation is controlled entirely by the pendulum. 
Thus the time during which the battery is kept closed 
through the resistance E may be made very short, and 
the condenser may be charged and discharged during this 
short interval of time. By this means polarization is 
reduced to a minimum and uniformity is secured. 




Fig. 45. 



108 



ELECTRICAL MEASUREMENTS. 



The connections for making a measurement of internal 
resistance are shown in Fig. 46. The pendulum is sup- 
posed to swing from left to right. When it strikes the 
lever or detent of key Ki contact is made on the upper 
point, and this closes the battery circuit through a known 
resistance B. The overturning of the lever key K. puts 




Fig. 46. 



the two terminals of tlie battery in connection with the 
condenser C. Wlien the pendulum reaches K:^ and 
overturns its detent IcA^er, the battery is removed from 
the condenser, and contact on the upper point causes a 
discharge through the galvanometer G. Finally, on 
passing K^ the pendulum operates this key and opens 
the battery circuit. To charge the condenser with the 
total E.M.F. of the battery, it is only necessary to leave 



RESISTANCE. 109 

the levers of Ki and K^ thrown forward. The circnit 
then remains open. After each reading the pendulum 
is brought back to the detent at the left, and the levers 
are then set up in the order in which they are thrown 
over by the pendulum. 

It will be observed that the battery circuit is open 
when the levers of keys K^ and ^4 are both up, and 
when they are both thrown over as well. This arrange- 
ment may be reversed so that the circuit is closed under 
the same circumstances, and is open only during the 
interval required for the pendulum to pass from K^ to 
K^ . This last arrangement is useful in getting the total 
E.M.F. of a cell while under test for polarization. The 
condenser is then charged and discharged while the 
battery circuit is open, and the recovery from polariza- 
tion will be negligible during this short interval. It is 
essential that the platinum contacts of the keys should 
be kept strictly clean. 

59. Resistance of Electrolytes — First Method. — 
All conducting liquids are electrolytes, except mercury 
and molten metals ; that is, the passage of a current 
through them is accompanied by the decomposition 
of the liquid conductor. If the rate of decomposition 
exceeds the rate of diffusion of the ions or products of 
the electrolysis, so that they accumulate on the elec- 
trodes, the result is a counter E.M.F. of polarization. 
This E.M.F. interferes with the measurement of electro- 
lytic resistances by the most simple means. The most 
usual method of annulling its effect is to employ rapid 
reversals of current or an alternating current of hip^h 
frequency. 

For this purpose a double commutator on one shaft is 



110 



ELECTRICAL MEASUREMENTS. 



appKcable. The shaft should be capable of rapid rota- 
tion by means of a crank and a train of gears. One 
commutator is included in the battery circuit and the 
other in that of the galvanometer. They should be set 
so that the current is reversed through the liquid at the 
same time that the galvanometer is commuted. The 
current reversals are supposed to be so frequent that 
polarization is annulled. The apparatus is shown in 
Fig. 47. 




Fig. 47. 



For the purpose of relative measurement of resistance 
or conductivity, comparison or standard solutions are 
needed. The following are recommended by F. Kohl- 
rausch ^ as good conducting solutions, having a conduc- 
tivity denoted by k at the temperature of t degrees C: 

NaCl, 26.-4 per cent, sp. gr. 1.201. 
h = 2015 X 10 -^^ + 45 X 10 -' G - 18)- 
MgSO^^ IT. 3 per cent, sp. gr. 1.187. 
y^ = 460 X 10-^ + 12 X 10"^ (t- 18). 

1 Wied. Ann. II., p. 653, 1880; Fh2/s. Jfeas., p. 320. 



BESISTANCE. 



Ill 




Fig. 48. 



These conductivities are relative compared with mer- 
cury at 0° C. But the specific 
conductivity of mercury is 1063 
X 10 ~^ C.G.S. units. Hence the 
conductivity of the above solu- 
tions in C.G.S. units may be 
found by multiplying the value 
of k by 1063 X 10 -\ 

To measure the conductivity of 
any liquid one of the standard 
solutions is iirst placed in the 
appropriate vessel (Fig. 48), de- 
signed by Kohlrausch. It is well 
to be provided with several of 

these vessels, with connecting tubes of different cross- 
section, adapted to 
liquids of different 
conductivity. 

The electrodes are 
platinized platinum, 
with their lower sur- 
faces convex. Let 
this liquid resist- 
ance be connected in 
one of the arms of 
the bridge, as Mi 
(Fig. 49), and let 
i^2, Ms J and Mi be 
non-inductive resist- 
ances. The contin- 
uous lines indicate 
permanent connec- 
tions inside the com- 
mutator box, the 




Fig. 49. 



112 ELECTEICAL MEASUREMENTS. 

dotted lines temporary connections outside. Then if the 
commutator is rapidly rotated the circuit through the 
galvanometer is reversed simultaneously with that 
through the battery and resistances. Hence the cur- 
rents through the galvanometer are rendered uni- 
directional. 

The resistances are then adjusted to balance, and the 
same relation subsists between them as in the case of 
steady currents. Next, fill the vessel with the electro- 
lyte to be measured and balance as before. The ratio of 
the two resistances will be the relative resistance of the 
two liquids, and their conductivities will be inversely as 
these resistances. 

Example. 

Standcard solution: NaCl, spec. grav. 1.201 at 18° C. 

Let kx equal the conductivity to be measured. The electrolyte 
was jjlaced in one arm of the bridge, and two incandescent lamps 
in another. Two resistance boxes, A and B, were in the other 
arms. Call the resistance of the lamps R. Then if r and r' are 
the resistances of the two solutions, 

r^R^\ r' = R^. 
B B' 



Whence h 


j,A B' 


Observation!^ : 




With standard solution. 




A B 


A 
B 


290 2000 


.1450 


2G8 1800 


.1461 


219 1500 


.1460 


Temperature 18.8° ; k = 


2180 X 10"' 



Mean. 
.1457 



BESISTANCK 



113 



Electrolyte. 


Temp. 


Spec. Gray. 


A' 


B' 


AB' 
BA' 


Conductivity. 


ZnSOi 


17.80 


1.0502 at 18.30 


1470 
1754 
2053 


1000 
1200 
1400 


.0992 


216 X 10-^^ 


ZnSOi 


18." 


1.2563 at 18.2» 


737 
1099 
1484 


1000 
1500 
2000 


.197 


430 X 10-^^ 


CuSOi 


17.8° 


1.0317 at 18.8° 


1840 
1377 
1155 


800 
600 
500 


.0633 


138 X 10-^^ 



60. Resistance of Electrolytes — Second Method. 
— Instead of a double commutator and a galvanometer, 
an induction coil or 
a sine inductor and 
an electrodynamome- 
ter (Art. 67) may be 
employed. This is the 
method of Kohlrausch. 
If the induction coil is 
used it should be one 
with a solid iron core, 
to avoid the great 
difference in the value 
of the direct and in- 
verse currents due to 



a wire core. 




Let I be the induc- 
tor (Fig. 50), U the ^'^- '°- 
electrodynamometer, M a post-office bridge, and V the 
electrolyte. The electrolytic resistance is one arm of 
the bridge. Tlie fixed coil of the electrodynamometer 
is in series with the main current, while the movable coil 



Professor Daniel, Physical Rev., Vol. I., No. 4, p. 241. 



114 



ELECTRICAL MEASUREMENTS. 



is connected in place of the galvanometer to the two 
ends of the proportional coils, a and b. By this means 
resistances can be measured to several significant figures. 
The sensibility is increased by increasing the current and 
shunting the bridge by a suitable resistance c d. 

The sine inductor may be used in place of the induc- 
tion coil. It may consist of a stationary Gramme ring, 




Fig. 51. 



inside of which rotates a two-pole field-magnet. Con- 
nection is made with the wire of the ring at two oppo- 
site points. This constitutes a simple alternating current 
generator. It may be driven by a direct current motor. 
If conductors are led off from four equidistant points 
on the ring, each pair of conductors, 180° apart, compose 
an alternating current circuit, and the generator is then 
two-phased (Fig. 51). 



BESISTANCE. 



115 



Example. 

Source of current, the sine inductor. E.M.F., 10 volts. 

The electrodynamometer contained two fixed coils. These 
were joined in parallel with one another, and the whole in par- 
allel with a Wheatstone's bridge. The movable coil was con- 
nected to the two ends of the proportional coils of the bridge. 

Standard solution : NaCl, spec, grav, 1.201 at 18° C. 

Observations : 

With standard solution, r = 41.47 ohms. 

Temperature, 24.4° C. ; k = 2410 X 10-^^ 



Electrolyte. 


Temp. 


Spec. Grrav. 


r' 


r' 


Conductivity. 


ZnSOi 
OuSOa 


24.1 

23.0 


1.0502 at 18° 
1.0317 at 18.8» 


444.1 
662 


.09338 
.06325 


225 X 10-1^' 
152 X 10-J3 



The difficulty in the way of effecting a balance arises 
from the E.M.F. introduced by capacity and induction. 

Chaperon has found that the static capacity of coils 
with "bifilar" winding of many turns produces a greater 
disturbance than the self-induction. To avoid this he 
winds the two wires, not side by side, but in alternate 
layers. It is better to wind in one direction only, and 
to bring each wire back parallel to the axis of the spool. 



61. Resistance of Electrolytes — Third Method. — 
Professors Ayrton and Perry have proposed a method 
which does not require the prevention of polarization. 
A current is passed through the solution between two 
plates of platinum, P, P (Fig. 52), till it acquires a 
constant value. Two platinum wires, w^ w^ are sealed 
into glass tubes and held rigidly in a fixed position 
between the platinum plates. The current is brought 



116 



ELECTBICAL ME A S UREMENTS. 



to some definite value and measured by an electrodyna- 
mometer or other current-measuring device. 

The potential difference be- 
tween the platinum wires is then 
measured by an electrometer 
or static voltmeter, E (Art. 95). 
An observation is first made 
with a standard solution and 
then \vith the electrol^^te to be 
measured, the current being 
brought to the same value each 
time. Then the resistances of 
the two liquids are proportional 
to the P.D.'s between the plati- 
num wires in the two cases. 
These P.D.'s can be in arbitrary 
units, since it is necessary to 
know their ratio only. 




Fig. 52. 



Example. _ 

Standard solution: NaCl, spec. grav. 1.201 at 18° C. 

Let k^ = the conductivity to be measured. Let d and di be the 
deflections of the electrometer with the standard solution and the 
unknown respectively. 



Then 



Deflection with standard NaCl solution . . . = 4.3 

Temperature of solution = 20.1° C. 

Specific conductivity (k) = 2243 X 10-^^ 

Observations : 



Electrolyte. 


Temp. 


Spec. Grav. 


Deflection. 


(1 
di 


Conductivity. 


ZnSOi 
CuSOi 


19" 

18.5° 


1.0502 at 18.3» 
1.0319 at 18.8° 


46.2 
70.75 


.093 
.062 


208.6 X 10-13 
139 X 10-13 



EESISTANCE. 117 

A serious objection to the method is the change in the 
density of the solution due to electrolysis. The deposit 
of zinc or copper on the platinum electrode reduces the 
density and the conductivity of the solution. 

The temperature coefficient of the ZnSO^ at 18° is 
0.0226, and of the OuSO,. 0.0215. When corrected for 
temperature, the three results for CuSO^ agree quite 
closely. The last two determinations of the ZnSO^ also 
agree fairly well, but the first is considerably higher. 



118 ELECTRICAL MEASUREMENTS. 



CHAPTER III. 

MEASUREMENT OF CURRENT. 

62. The Tangent Galvanometer. — The tangent 
galvanometer has lost most of its former importance, 
bnt it is useful in a laboratory, and will be described 
because of its historical importance, if for no other 
reason. In its simplest form a tangent galvanometer 
consists of a circular conductor, supported vertically in 
the magnetic meridian, and having at its centre a mag- 
netized needle capable of turning around a vertical axis. 
The length of the needle must be short in comparison 
with the radius of the coil. Tliis is essential, so that 
when the needle is deflected by a current the movement 
shall not place the poles in, a field of magnetic strength 
different from that at the centre of the coil. A small 
deflection of a long needle would move its poles from 
the uniform field in the plane of the coil to a relatively 
weaker one on either side of this plane. The lines of 
force due to a current circulating around a circular coil, 
or the lines along which a magnetic pole is urged, 
coincide with the axis of the coil at its centre. Near 
the centre they are very nearly parallel lines. If, there* 
fore, a short needle, in length from one-twelfth to one- 
tenth the diameter of the coil, has its magnetic axis in 
the plane of the coil when no current is passing, then 
when it is deflected by a current, the direction of the 
deflecting force acting on its poles in the new position 



MEASUBEMENT OF CUBBEJS'T. 119 

will be perpendicular to the plane of the coil, and the 
orienting force due to the earth's magnetism Avill be 
exactly at right angles to the deflecting force. 

Let JVS (Fig. 53) be the magnetic meridian, and let 
the plane of the coil be in it, with its centre at 0. Let 
the needle be deflected through an angle ^. Then two 
forces act upon each pole. One is SfSm^ \ due to the 
horizontal component of the earth's 
field ; the other, due to the current 7, is 
at right angles to the plane of the coil, 

and equals , where r is the mean 

radius of the coil, consisting of a single 

turn of the conductor, and ni is the 

strength of pole of the needle. For 

equilibrium the moments of these two 

forces, or the moments of the two ^'^' ^^' 

couples, due to the pairs of equal forces acting on the 

two ends of the needle, must be equal to each other. 

The moment of the orienting magnetic force due to the 

earth's field is &6'ml sin ^, where I is the half length of 

the needle. The moment of the deflecting force is 

L cos c7. Hence, 

r 

?^^^ cos d = &S^l sin ^. ^ co^ a r H 5H. ^ 

The m and I cancel out. The deflection is therefore 
independent of the strength of pole; but the length 
is limited for a reason already given. 

From this equation 

1= ml- tan e. 




120 ELECTRICAL MEASUREMENTS. 

For n turns of wire, where n is only a very small 
number, and where the n turns may be considered 
coincident, 

J=a?~tan(9. 

TT 277"?? 

The fraction ^^, or , is called the constant of the 

r r 

galvanometer. 

It depends upon the dimensions and the number of 
turns of wire in the coil, and equals the strength of field 
produced at the centre by unit current flowing through 
the coil. If this constant be denoted by (x, then 

J=^%.an^ (1) 

G- 

The equation may be written simply 

J=^tan^ (2) 

The current is measured in C.G.S. units. The num- 
ber of amperes is 10 times as great. When the constant 
of the galvanometer is determined from its dimensions, 
equation (1) must be used ; when it is determined by silver 
or copper electrolysis, equation (2) is more convenient. 

63. Influence of Errors of Observation. — The 

effect of an error in reading the deflection of the needle 
of a tangent galvanometer will be least relative to I 
when 6 = 45°. This may be demonstrated by applying 
the formula of Article 36 to the tangent galvanometer, 
the equation of which is 

1= A tan 0. 

Then 

^=4r 



MEASUREMENT OF CURRENT. 121 

But 

dl A 



dd cos- e 

Therefore 



F=^A 



J f cos 



cos^ 6 cos- sin 6 



by the substitution of ^ for the constant A. 

^ tan d 

But 

/cos^ _ / _ 2/ > 

cos^ ^ sin 6 sin ^ cos ^ sin 2^ 

XT 

Hence, - will be a minimum for any error of ob- 

2f 
servation / when — ^^ is a minimum, or when sin 26 is 
-^ sin 2(9 

a maximum. The sine of an angle is a maximum when 

the angle equals 90° ; and sin '26 is therefore a maximum 

when 6* =.45°. 



64. Plotting- Currents measured by a Tangent 
Galvanometer. — Let R equal the resistances in the 
circuit of the galvanometer except that of the battery, 
and let r be the resistance of the battery. Then by 
Ohm's law 

B + r 

If now a constant E.M.F. be employed, and if the in- 
ternal resistance of the battery does not change, 

I oc tan 6 oo -- — oc ^ . 

R-\-r cot 6 

Hence cot 6 cc M -j- r. 



122 



ELECTBICAL MEASUREMENTS. 



If, therefore, we plot a curve with the tangents of the 
several observed deflections as ordinates and the corre- 
sponding resistances M as abscissas, we shall obtain the 
curve J (Fig. 54), which is an hyperbola. Plotting cotan- 
gents of as ordinates and resistances M as abscissas, on 
the other hand, gives the straight line II. The two 



2.2 














/ 


















2.0 




\' 








.,/ 


II 


















1.8 




\ 






^C 


f 




















^1.6 




\ 






# 























ii.4 




\ 




/ 


y 




















6 

•S1.2 




\ 


\, 


/ 


















1 




sl.O 






V 


/ 
























^0.8 






/ 


\, 
























0.6 




/ 




\ 


\ 






















0.4 




/ 






V 


\ 


^ 


















0.2 


/ 














"^ 







Tan^ 


7e7its 








/ 


/ 



1 





2 





3 


0°^ 


-4 


[) 


5 





6 





7 


3 



Fig. 54. 

curves intersect at a point corresponding to a deflection 
of 45°. The cotangents' line does not intersect the axis 
of resistances at the origin, but at a distance to the left 
equal to the internal resistance of the battery r. 



65. To find the Magnetic Field at any Point on the 
Axis of the Coil. — Let DJS (Fig. 56^ be the trace of 
the plane of the coil, and let OC he its axis. It is 



MEASUREMENT OF CURRENT. 



123 



required to fiiid the deflecting force at any point A on 
the axis. Let AB represent the force on unit pole at A 
due to the current in a small element ds of the circle 

at U. It will equal —^, and its direction will be per- 
pendicular to UA in the plane 
of the paper. The deflecting 
force at a point is always per- 
pendicular to a plane containing 
the element of the conductor 
and a line drawn from the 
middle point of the element to 
the given point. The effective component AC is 




Ids 



sm a 



Ids 

IF 



Irds 



(r' + d'^i 



Hence for one turn of wire 



8- = 



2SI 



^2)1 



where S is the area of the circle 7rr\ This expression 
represents the total force, since the components CB 
balance one another all around the circle. Each element 
of the circle has a symmetrical one at the other extremit}' 
of a diameter through it, and the component at right 
angles to the axis of the coil, due to this symmetrical 
component, is equal and opposite to CB. 

If a magnet pole of strength m is placed at A, then 
the force on it perpendicular to the plane of the ring is 
Efm, while the horizontal force due to the earth's magnet- 
ism is cfSm. Hence, as in the tangent galvanometer, 

g^m cos = g^m sin ^, 
or 8^ = Ef6 tan 6. 



124 

Therefore 
or 



ELECTRICAL MEASUREMENTS. 



(^2 ^ ^2>^f 



K 



(r^ + d'}i 



tan 0. 



jBT is a constant if I and &S are constant. This form 
of the equation is convenient for use in the experimental 
proof of the relation between d and 0. 

Example. 

Place the circular coil (Fig. 56) in the magnetic meridian and 




Fig. 56. 



set the compass box at the centre of the coil. Pass a cmTent 
through the coil from a constant source, and of such strength as 
to give a deflection of about 45°. By means of a commutator 
take deflections in both directions. Repeat the observations with 
the compass box at diff'erent distances from the centre of the coil. 
Measure the mean radius of the coil with calipers, if it is not 
known. Finally, compare tangents of the mean deflections with 
the values derived from the preceding equation and plot.^ 



1 Stewart & Gee's Frac. Phys., Part II., p. 321. 



MEASUREMENT OF CURRENT. 



125 



The following are the details of an experiment 
r 12.45 cms. 



(1) 

Distances 

X 

in cms. 


(•2, 

Mean 

Deflection. 


(3) 
Tangent 

Deflection. 


<^ 


(5) 

(3) H- (4) 

or K. 


(6) 
A' ^2 

3 

(r2 + x2)2 


3 





43° 21' 


0.9440 


0,08032 


11.753 


0.9394 


1 


43.3 


0.9341 


0.07955 


11.742 


0.9304 


2 


42.12 


0.9067 


0.07731 


11.729 


0.9042 


3 


40.51 


0.8647 


0.07380 


11.717 


0.8631 


4 


39.9 


0.8141 


0.06932 


11.745 


0.8107 


5 


37.6 


0.7563 


0.06418 


11.783 


0.7506 


6 


34.33 


0.6886 


0.05872 


11.726 


0.6858 


7 


32.3 


0.6261 


0.05320 


11.769 


0.6222 


8 


29.12 


0.5589 


0.04783 


11.685 


0.5593 


9 


26.30 


0.4986 


0.04275 


11.662 


0.5000 


10 


23.34 


0.4431 


0.03807 


11.642 


0.4452 


11 


21.27 


0.3929 


0.03381 


11.623 


0.3953 


12 


19.15 


0.3492 


0.02998 


11.648 


0.3506 


13 


17.12 


0.3096 


0.02658 


11.647 


0.3108 


14 


15.18 


0.2736 


0.02357 


11.606 


0.2757 


15 


13.42 


0.2438 


0.02093 


11.650 


0.2447 


16 


12.15 


0.2171 


0.01860 


11.671 


0.2176 


17 


10.57 


0.1935 


0.01655 


11.691 


0.1935 


18 


9.48 


0.1729 


0.01479 


11.682 


0.1729 


19 


8.48 


0.1548 


0.01322 


11.706 


0.1547 


20 


7.54 


0.1388 


0.01185 


11.705 


0.1386 



The mean value of ^is 11.695. From this value and 
from column (4) column (6) is calculated. The curve 
(Fig. 57) represents the observed values of the tangents, 
distances x being plotted as abscissas. The curve of the 
theoretical values of the tangents in column (6) falls so 



126 



ELECTRICAL MEASUREMENTS. 



near this one that it cannot be plotted separately. The 
greatest difference between the observed and computed 
values of the tangents is only three-fourths per cent ; 
most of these differences are only a small fraction of one 
per cent. 



< 

90 
80 
70 
60 
50 




S 


^, 












































N 


\ 


>> 












































N 


\ 














































\ 


^ 












































\ 


\ 














































\ 


\ 
























30 
20 
























s 


s 


i 












































N 


s 


s 


^, 
















































S 


*^ 


) 








L 


































L_ 









10 
Fig. 57. 



20 



66. The Cosine Galvanometer. — The cosine gal- 
vanometer is made so that the coil may rotate about its 
horizontal diameter. Fig. 58 is a vertical section through 
the coil. The axis of rotation, which lies in the mag- 
netic meridian, is perpendicular to the paper through 0. 
The plane of the coil has been rotated over through the 



MEASUREMENT OF CURRENT. 127 

angle (/). Then the whole force on the magnet pole due 
to the current through a single turn of conductor is in 
the direction 0(7, perpendicular to the 
plane of the coil. The effective compo- 
nent moving the needle is OD, and 

OD — cos (/). 

Placing the moments of the two forces a 
acting on the needle equal to each other, ^'°- ^s- 

as in the case of the tangent galvanometer, we have 

BS'ml sin 6 =^- I cos 6 cos <^, 

r 

or 

r tan 6 



I=&6 



lir cos (^ 



For a given deflection of the needle the current 
is inversely proportional to the cosine of the angle 
which the plane of the coil makes with the vertical. 
By this means the range of the galvanometer is greatly 
increased. 



67. The Siemens Blectrodynamoraeter. — An elec- 
trodynamometer consists of two coils with their mag- 
netic axes at right angles, one of them fixed and the 
other movable about a vertical axis through its plane. 
The motion of the movable coil is produced by the 
electrodynamic action between the convolutions of 
the two coils. The current flows through the two in 
series. 



128 



ELECTRICAL MEASUBEMENTS. 



Let AB (Fig. 59) be a single convolution of the fixed 
coil and CD the suspended movable coil. The movable 
coil consists of only one turn, or at least a very limited 
number, according to the current which the instrument is 
designed to measiu^e. A large cur- 
rent means a healy conductor and 
a single turn, since it would be im- 
practicable to support several tui^ns. 
The instruments for smaller cur- 
rents may have several turns in the 
movable coil. It will be seen that 
the movable conductor is subjected 
to a system of forces all tending to 
tui'u it in the same direction. It is 
suspended by means of silk threads 
or on a point resting in a jewel ; 
and a carefully wound helix is con- 
nected rigidly \\ith it and with the 
torsion head T above. Fig. 60 
shows the complete instrument. 

When the coil turns by passing 
current through it, the turning of 
the torsion head brings it back 
again to the zero or initial position. 
Thus the couple due to the electrodynamic action is 
offset by the couple of torsion of the helix connected 
with the torsion head. This torsion couple is therefore 
employed to measure the current. Now, the couj)le of 
torsion is proportional to the angle of torsion by Hooke's 
law, the forces of restitution which are called into action 
by any distortion within elastic limits being propor- 
tional to the distortion itself. But the electrodynamic 
action is proportional to the square of the cui^rent, since 




Fig. 59. 



MEASUBEMENT OF CUEBENT. 



129 



the two coils are in series. Hence the square of the 
current is proportional to the twist of the counteracting 
helix. 

We may accordingly write 



or 



P = A' J), 



I=A\/D, 



as the equation connecting the current with the twist 
of the torsion helix. J. is a 
constant depending upon the 
windings, the torsion of the sus- 
pending spring, etc. This is the 
common equation of a parabola. 
Hence if currents and twist be 
plotted as coordinates, the result- 
ing curve will be parabolic. 

Two fixed coils are commonly 
employed, one of fine wire and 
the other of coarse wire. 
One end of each is 
connected to a binding- 
post on the base of the 
instrument. The other 
terminals are connected 
to the upper mercury 
cup at a (Fig. 59), into 
which dips one end of the movable coil, the other end 
dipping into another mercury cup at />, from which a 
conductor leads to a third binding-post. Hence, whether 
the current enters by the one fixed coil or the other, it 
passes out through the suspended coil and the third post. 
Since the direction of the deflection depends upon the 




130 ELECTRICAL MEASUREMENTS. 

manner in which the coils are connected, and not upon 
the direction of the current, the electrodjaiamometer is 
applicable to the measurement of alternating currents. 
Its period of swing must, however, be long in com- 
parison with the period of alternation of the current. 
It then becomes an integrating device, and integrates 
the values of the squares of the current for successive 
equal time-intervals. The result is, therefore, the square 
root of the mean square of the current. 

68. The Equation of the BlectrodynarQonieter as 
affected by the Earth's Field. — - AVhen only small 
currents are employed with a sensitive electrodynamome- 
ter, the effect of the earth's directive force on the sus- 
pended coil, considered as a magnetic shell, must be 
taken into account. This force is proportional to the 
fost power of the current, while the deflecting force due 
to the mutual action of the coils is proportional to the 
square of the current. If, therefore, the instrument is 
set up with the plane of the suspended coil and the axis 
of the fixed coils in the magnetic meridian, the fixed 
coils being of such dimensions as to produce a sensibly 
uniform magnetic field in the region of Jhe suspended 
coil, we shall have for the equation of equilibrium 

ar COS e + bl cos 6 = 00, 
or 

ar+bI=c-L, .... (1) 
cos 6 

in which a is a constant depending on the windings and 
dimensions, h one depending on the number of turns 
and the area of the suspended coil, as well as on the 
earth's horizontal field d€n c the couple of torsion for a 



MEASUREMENT OF CUBBENT. 131 

unit angle, and 6 the deflection. The current is here 
supposed to be in the direction in which the earth's 
magnetic force and the electrodynamic action between 
the coils act together. 

If the current be reversed the dynamic action between 
the coils turns the suspended coil the same way round, 
but the direction of the couple due to the earth's field 
is reversed. Therefore, the deflecting couple is due to 
the difference of the two forces, and for the same deflec- 
tion as before the current must be greater. Let it be 
n times as great. Then we may write 

awT- — hnl = c , . . . (2) 

cos^ 

anl'- -\- hnl — nc ^. . . . (3) 

cos 6/ 

Multiplying equation (1) by ??, we have equation (3) ; 
adding and dividing by (tz + 1), we have 

anP — c . 

cos 6 

It follows, therefore, that if the earth's influence were 
eliminated, the same deflection 6 would be given by a 
current equal to Isjn^ numerically a mean proportion 
between the two oppositely directed currents required to 
produce the same deflection. 

For small angular displacements equation (1) may be 
written with sufficient approximation, 

aP + hI=ccl, (4) 

where d is the deflection in millimetres observed by the 
usual telescope and scale method, and c is dependent on 
the distance of the scale from tlie electrodvnamometer. 



132 ELECTRICAL MEASUREMENTS. 

Equation (4) is the equation of a parabola referred to 
axes 'parallel to those of the equal parabola whose equa- 
tion is 

a 

The following equation was derived from a sensitive 
instrument in our laboratory: 

P - 0.8427= 0.0298c^. ... (5) 

If the current through the suspended coil alone is 
reversed, we obtain 

P + 0.842Jr= _ 0.0298t^. ... (6) 

If the observations are plotted with deflections as 
abscissas and currents as ordinates, the full line parabola 
passing through the origin is obtained (Fig. 61). 

For alternating currents the term containing the first 
power of I in equation (5) vanishes, and we have 

7= \/Oj0298^ = 0.1726 V^. . . (7) 

This equation represents the same parabola as that of 
equation (5), but shifted, as shown in the dotted curve in 
the figure, so as to have its vertex at the origin. It is 
the equation for alternating currents in which the earth's 
field plays no part. For direct currents the instrument 
should be set up with the plane of the movable coil at 
right angles to the magnetic meridian. 

69. The Wattmeter. — The electrodynamometer may 
be made to measure the power expended in any part of 
a circuit. The integrated product of the current and 
the corresponding pressure at the terminals of the circuit 
is the mean power expended in it. If the whole current 



MEASUREMENT OF CURRENT. 



133 





I 




1 




^"^ 


X 


Milliat 


\,peres' 


JO 












/f 


^^\^ 


\ 


\ 














/ 






\ 


\ 














'' i 


^o 




\ 
\ 


\ 














I P 


OT 




\ 


























. 










/ 










\ 
\ 
\ 


\ 










— ^ 




— s 






\ 


v 










/ 

/ / 
/ / 










\ 
\ 

\ 
\ 
\ 
\ 


1 








/ 


/ 




O' 








\ 
\ 


\ 






I 


/ 












\ 
\ 


\ 






1 
1 


/ 




s 








\ 
\ 
\ 

\ 


\ 






1 
1 
1 


' 












\ 
\ 

1 

\ 


\ 






' 






JS 








1 
\ 






; 
1 


/ 






cS 










\ 


\ 


! 


/ 






o 


1 








\ 
\ 

\ 
\ 
\ 




I 
1 


f 








1 








\ 


\ 


1 


/ 






>-x 










\ 


\ 


1 


1 






<jy 












\ 










i 












\ 



Fig. 61. 



134 



ELECTRICAL MEASUREMENTS. 



is carried through the fixed coil of the electrodynamom- 
eter, and the movable coil is connected as a shunt to the 
resistance on which the power to be measured is ex- 
pended, so as to serve as a pressure coil, with the neces- 
sary resistance in series with it, the instrument then 
becomes a wattmeter, and may be calibrated to read 
in watts. 

Fig. 62 is the 
Weston wattmeter, 
which is graduated 
to read directly in 
watts. Fig. 62a 
is a diagram of the 
internal connec- 
tions. The trans- 
lating device, such 
as a lamp, is con- 
nected across the 
mains from to 
D. A and B are 
the terminals of 
the series or field 
coiL and ah those of the pressure coil. It will be seen 
that the pressure circuit through the movable coil is 
carried roimd the field coil also. This is for the pui'pose 
of compensating for the current through the pressure 
circuit, since this current also traverses the series coil. 
The connections are so made that the cui'rents through 
this compensating winding and the field coil flow in 
opposite directions. The reading is thus diminished to 
such an extent as to compensate for the energy required 
to operate the instrument. 

The independent binding-230st I is employed in con- 




Fig 62. 



MEASUREMENT OF CURRENT. 



135 



nection with b when the instrument is used with two 
independent circuits, or when it is calibrated by means 
of two separate cur- 
rents. The compen- c ^^^ d 

sating winding is ^ 
then cut out and an 
equivalent resist- 
ance is substituted. 



Wvwvvvv^ 




70. Thed'Arson- 
val Galvanometer. 
— The d' Arson val 
galvanometer may 
be considered as an 
electrodynamom eter 
in which the fixed 
coil is replaced by a 
permanent magnet; 
or it may be looked 
upon as a galvanom- 
eter in which the 
magnet is fixed and 
the coil is movable, 
instead of the converse arrangement of the tangent gal- 
vanometer. Since the action and reaction are equal 
between a coil and a magnet, it is immaterial from a 
magnetic point of view whether the one is made movable 
or the other. 

The great advantage of the d' Arson val type of gal- 
vanometer is that it has a strong magnetic field only 
slightly affected by the earth's magnetism, or by iron or 
other magnetic matter in its vicinity. It is also ex- 
tremely '^ dead beat " under certain conditions. Further- 



Fis. 62a. 



136 



ELECTRICAL ME AS UBEMENTS. 



more, by properly shaping the pole pieces of the 
permanent magnet, the deflections may be made strictly 
proportional to the current. The Weston instruments 
for direct currents are a modification of the galvanometer 
of d'Arsonval, and both operate on the same principle 
as Lord Kehdn's Siphon Recorder for submarine teleg- 
raphy, which preceded both of them. 

In the earlier instruments 
of this design the coil had 
a large area, and a soft 
iron core was inserted to 
strengthen the field. This 
arrangement is still retained 
in the Weston instruments. 
But A}i*ton has pointed 
out ^ that galvanometei^ of 
the d'Arsonval type should 
not have a soft iron core, 
and that the coil should 
be long and thin. 

Fig. 63 is a d'ArsouA-al 
galvanometer of ordinary 
pattern. The current is 
led in through the spring 
and attached wire at the 
bottom, thence through the coil, and out by the sus- 
pending wire and the supporting standard. The field- 
magnet is a compound one supported vertically. Within 
the coil is a soft iron core supported from the rear. 
The coil turns in the narrow intense field between the 
poles of the magnet and the iron core. When the 
external resistance is not large, the induced currents on 

1 <* Galvanometers," Fhil. Mag., July, 1890, p. 58. 




Fig. 63, 



MEASUREMENT OF CURRENT. 



137 



closing the current, with the coil in motion, quickly 
bring it to rest. 

The coil in the Weston instrument (Fig. 64) is con- 
trolled by two helical springs which also serve as con- 
ductors to connect the coil with the external circuit. A 




Fig. 64. 



portion of one pole is shown cut away in the figure. 
The pivots rest in jewels, and a long aluminium pointer 
is attached to the coil and traverses a scale of equal parts 
not shown. In the voltmeter a large resistance is put 
in series with the movable coil. In the ammeter for 
large currents the movable coil is connected as a shunt 
to the main conductor in the instrument. 



138 



ELECTRICAL MEASUREMENTS. 



The Ayrton-Mather pattern of this galvanometer (Fig. 

Qb') has a single ring-magnet with a narrow division at 

one point. In the opening is 
placed the tube containing 
the long narrow coil without 
any iron core. This coil is 
suspended by a thin wire, and 
has a fine helix at the bottom 
for a conductor. Its plane 
must be parallel to the lines 
of force in the narrow gap in 
which it hangs. If quick 
'^" ' damping is desired, the coil 

is enclosed in a thin silver or aluminium tube. 




,9 



iEi-.:xm 



71. The Best Shape for the Section of a Coil. — The 
best shape for the section of the coil of a d'Arsonval 
galvanometer perpendicular to the axis about which it 
turns has been determined by Mather.^ 

His paper deals with coils sus- 
pended in a uniform field, but similar 
reasoning applies to instruments in — —- 
which the field is not uniform. -~— - 

Let the field be of strength cfS^ and 
let P (Fig. QQ^ be an element of the 
section of the coil turning about an Fig. 66. 

axis through A perpendicular to the 
plane of the element, and / the current density per unit 
area. Then the deflecting moment exerted on unit 
length, measured perpendicular to the paper, and of 
cross-section a, is 

BSlar sin 6. 



Phil. Mag., Vol. 39, p. 434, May, 1890. 



MEASUREMENT OF CUBBENT. 139 

The moment of inertia of the element about A will be 

where d is the density, or mass per unit cube. 

In ordinary instruments it is inconvenient to have the 
period of oscillation long, but for a constant period the 
controlling moment at unit angle must be proportional to 
the moment of inertia ; hence the problem is to find the 
shape of the section such that the total deflecting 
moment for a given moment of inertia shall be a 
maximum. 

If the magnetic moment of a spiral be made greater 
by increasing its radius, the moment of inertia will be 
increased in a greater ratio, and thus the period of free 
vibration of the coil will be increased. But this period 
is limited by practical considerations. We have, there- 
fore, to consider the form, so that for a given moment of 
inertia there may be a maximum magnetic moment ; or, 
what amounts to the same thing, for a given magnetic 
moment the coil may have a minimum moment of 
inertia. 

The ratio of the magnetic or deflecting moment to the 
moment of inertia of the element considered is 

EfSIar sin 6 >_ sin 6>. 

ardr rd 

Since S^, J, and d may be considered constants, the prob- 
lem is to find the conditions making a maximum for 

every element of the coil. 

Consider the curve the polar equation to which is 

r =^c sin 0. 

For a given value of 4- c the equation represents two 




140 ELECTRICAL MEASUREMENTS. 

circles tangent to ^(7 at the point A (Fig. 67). The 
diameter of the circles is c. A family of such circles 
may be drawn with A as the com- 
mon point of tangency. If now we 
conceive a wire transferred from the 
surface of the circle to a point with- 
out, then the value of c for such 
outer point is greater, and conse- 
quently — ^ — is less than for a point 
on the circumference. If it is trans- 
ferred to a point inside the circle, the value of is 

greater. If, therefore, the cross-section of the coil be 
any circle, r = e sin 0, a diminution of the value of the 

expression ^^ — would be produced by transferring any 

portion of the wire within the circle to any unoccupied 
space outside ; that is, the ratio of the magnetic moment 
to the moment of inertia would be diminished. 

Also, since the horizontal portions of the coil, lying 
parallel with the field, contribute to the moment of 
inertia and not to the deflecting moment, the deflecting 
moment will be increased by making the coil long and 
narrow. The cross-section of the long narrow coil must, 
moreover, be two tangential circles, their point of tan- 
gency being as nearly as possible on the axis of- rotation 
of the coil.^ The problem in hand " resolves itself into 
fuiding the shape and position of an area having a given 
moment of inertia about a point in its plane such that 
the moment of the area about a coplanar line through 

1 Gi-ay's Absolute Measurements in Electricity and Magnetism, Vol. II., Part 
II., p. 380. 



MEASUREMENT OF CURBENT. 



141 



the point is a maximum. Taking the point as a pole, 
these conditions are 



while 



JJ r^ dr dO is a constant, 

Xf^'' sin dr dO is a maximum." 



72. The Kelvin Balances. — In the balances of Lord 
Kelvin the electrodynamic action between the fixed and 
movable coils is counterbalanced by adjustable weights 
or sliders instead of the torsion of a helical spring. 

The coils are ring-shaped and horizontal. The two 
movable rings E and F (Fig. 68) are attached to the ends 




Fig. 68. 

of a horizontal balance beam which is supported by two 
trunnions a and &, each hung by an elastic ligament of 
fine wires, through which the current passes into and out 
of the circuit of the movable rings. These rings are 
placed midway between two pairs of fixed rings, AB and 
61), which are connected as shown in the diagram, so 
that the movable ring on either side is attracted by one 
of the fixed rings and repelled by the other. When a 
current passes through the six coils in series, the beam 
tends to rise at F and sink at F. 



142 ELECTBICAL MEASUREMENTS. 

The balancing is performed by means of a weight, 
which slides on a horizontal graduated arm attached to 
the balance beam (Fig. 69). A trough is fixed to the 
right-hand end of the beam, and in it is placed a weight 
which counterpoises the sliding weight, shown near the 
centre of the beam, when it is at the zero of the scale 
and no current is passing through the balance. By this 
arrangement the range of movement of the slider is the 
entire length of the beam. These weights can be 
changed so as to vary the range of the balance. Pro- 
vision is made for the fine adjustment of the zero by 
means of a small metal flag, as in some chemical 
balances. A vertical scale and a horizontal pointer at 
each end of the balance arm determine the sighted zero 
position. T\^hen a cuiTent passes, the beam is brought 
back to the horizontal position by moving the sliding 
weight toward the right by means of a self-releasing pen- 
dant, hanging from a hook carried by a sliding platform, 
which is pulled in the two directions by two silk cords 
passing through holes to the outside of the glass case. 
The balance is shown in the figure with the glass case 
removed. Since the force is proportional to the product 
of the current in the fixed and movable coils, the current 
is proportional to the square root of the turning moment. 
Hence the four pairs of weights (slider and counter- 
poise) supplied with each instrument are adjusted in the 
ratios of 1 : 4 : 16 : 64, so that for the same division of 
either scale the second weight indicates twice the cur- 
rent of the fhst, the third twice that of the second, and the 
fourth twice that of the third. Of the two scales 
the upper fixed one, called the inspectional scale, gives 
the current approximately in decimal parts of an ampere ; 
but for more accurate reading the movable scale of equal 



MEASUBEMENT OF CURRENT. 



143 




144 ELECTRICAL MEASUREMENTS. 

parts must be read, and the current calculated by the aid 
of the table of doubled square roots (Appendix, Table 
VI.). Thus, for example, if the balancing point is 475 
on the scale of equal parts, the corresponding reading 
for the inspectional scale obtained from the table is 
43.59. 

There are several types of instruments made. The 
following table shows the value per division of the 
inspectional scale corresponding to each of the four pairs 
of weights for the centi-ampere, the deci-ampere, the 
deka-ampere, and the hekto-ampere balances : 

I. n. m. IV. 

Centi-amperes Deci-amperes Amperes Amperes 





per 
division. 


per 
di%'i3ion. 


per 
division. 


per 
division. 


1st pair of weights . , 
2d " - " . . 


. . 0.25 
. . 0.5 


0.25 

0.5 


0.25 
0.5 


1.5 
3.0 


3d " " " . , 


. . 1.0 


1.0 


1.0 


6.0 


4tli " " 


. 2.0 


2.0 


2.0 


12.0 



The useful range of each instrument is from 1 to 100 
of the smallest current for which its sensibility suffices. 
Thus the centi-ampere balance, illustrated in Fig. 69, 
has a range with the four weights from 1 to 100 centi- 
amperes, or from y^ to 1 ampere. 

Each balance is designed to carry 75 per cent of its 
maximum current continuously, and its maximum cur- 
rent long enough for standard comparisons. 

The centi-ampere balance, with a thermometer to test 
the temperature of its coils, and in the more recent 
instruments with platinoid resistances up to 1,600 ohms, 
serves to measure potential differences of from 10 to 400 
volts. The fu'st resistance of the series includes that 6f 
the balance, which is about 50 ohms. 



MEASUREMENT OF CUUBENT. 145 

Constants of Centi-ampere Balance as a Voltmeter. 

Resistance in Volts per division of 
Weight used. circuit. fixed scale. 

1st pair 400 1.0 

" " 800 2.0 

" " 1,200 3.0 

" " 1,600 4.0 

If the second pair of weights is used, the constants 
will be double those in the last column. 

For the highest accuracy corrections must be made 
for the temperature of the balance and of the auxiliary 
platinoid resistance. The correction for the copper 
resistance of the former is about 0.4 per cent per degree 
centigrade, and for the latter about 0.024 per cent. 

When the lowest potentials are measured the smallest 
platinoid resistance must be in the circuit ; and one or 
more of the others must be included in series with it, 
when the potential is so high as to give a larger current 
than can be measured by the lightest weight on the 
beam. 

73 . The Thomson Astatic Reflecting Galvanometer. 

— For the highest sensibility the requirements of a 
good galvanometer^ are : 

(a) An astatic magnetic system of small moment of 
inertia. 

(5) A variable magnetic control. 

(c) Four coils of nearly equal resistance. 

(d) High insulation and large resistance. 

Such an instrument is shown complete in Fig. 70. The 
coils are supported on grooved pillars for the purpose of 
increasing their insulation from the base. The binding- 
posts on the top are the terminals of vertical brass rods 
which screw into special lugs on the coil frames. They 



146 ELECTBICAL MEASUREMENTS. 

* are disconnected from tlie case when in use. The open- 




Fig. 70. 



y 



ing between the rods and the case can be closed by 



MEASUREMENT OF CURRENT. 



147 



rubber washers when the instrument is not in use. The 
control magnet on the vertical supporting rod is similar 
to the one on the tripod galvanometer of Fig. 8. The 
suspension is by means of a quartz fibre which is greatly 
superior to silk in strength, stability, uniformity, and 
smalhiess of torsion coefficient. 

Fig. 71 is a galvanometer of similar construction. It 
shows the two 
coils on one side 
swung open, ex- 
posing the as- 
tatic magnetic 
system. 

The magnetic 

system consists 

of two sets of 

minute magnets 

made of bits of 

fine watch- 

spring. Four or 

five of these are 

attached near 

the top of a thin 

aluminium wii'e 

with their north- 

seeking poles 

turned toward 

the north ; the 

same number are 

similarly attached at the bottom, but with the north- 
seeking poles turned toward the south. The first set is 

placed at the centre of the upper pair of coils, and the 

other set at the centre of the lower pair. Between them 




Fig. 71. 



148 ELECTBICAL MEASUREMENTS. 

a small round mirror is hung in a very light aluminium 
cradle. This is either plane or concave, according as it 
is desired to read the deflections with a telescope and 
scale or with a lamp and scale. 

If an incandescent lamp be available, by enclosing it 
in an appropriate case or hood, it may be used with a 
translucent scale, and may give sufficient light to read 
the deflections in a well-lighted room. 

The movable system weighs only a fraction of a 
gramme. The arm carrying the suspending fibre savings 
out so that the system is entirely free and can be readily 
examined or conveniently mounted. The contact be- 
tween the coils is automatic, and is made by means of 
platinized springs when the hinged face is closed. The 
use of flexible conductors is thus avoided. 

The control magnet M., of Fig. 71, is novel and con- 
venient. It not only turns around a vertical axis, but 
its effective magnetic moment can be varied by turning 
the milled head S. It consists of a permanent cylindrical 
magnet with threads cut on each end. On these threads 
turn two long nuts of soft iron which act as a magnetic 
shunt. They approach or recede from eacli other accord- 
ing as the magnet is turned by the milled head in the one 
direction or the other, since one thread is right hand and 
the other left. In this way the sensibility can be regu- 
lated with great exactness. The field produced by the 
control magnet at the needles is changed by the magnetic 
shunt instead of by changing the distance of the magnet 
from the suspended system. 

It is customary to give to the upper set of magnets a 
slightly greater magnetic moment than that of the lower 
set. The entire system then places itself in the mag- 
netic meridian, but with a very feeble directive force. 



MEASUREMENT OF CURRENT. 149 

The mirror is commonly attached so as to look toward 
the west when the galvanometer is adjusted. The 
aluminium disks at the needles are intended to produce 
air damping, and to aid in bringing the movable system 
more rapidly to rest after deflection. 

To adjust the galvanometer, proceed as follows : 
Place it on some fixed support, such as a pier with a 
stone top, or on a shelf attached to a brick wall. Turn 
the instrument till the plane of the coils is as nearly as 
may be in the magnetic meridian. Next level by means 
of levelling screws till the movable system hangs en- 
tirely free within the coils. In lifting, the system by 
the suspension pin, it should be raised very slowly and 
carefully till the needles are in the centres of the coils. 
They should then be entirely free, and the suspending 
fibre should be without torsion. The scale should then 
be placed at the proper distance from the galvanometer 
in the magnetic meridian, and horizontal. Next turn 
the control magnet till the plane of the mirror is in the 
magnetic meridian as nearly as possible. One can judge 
of this by looking into the mirror and getting an image 
of one's eye. Then move backward and observe if the 
line of sight is perpendicular to the face of the instru- 
ment. If not, adjust by turning the control magnet. 
Then make the height of the telescope and scale such 
that on looking directly along the tube of the telescope 
an image of the scale can be seen in the mirror. Focus 
the telescope and finally adjust the image by slightly 
changing the height of the scale, and by the altitude and 
azimuth screws on the telescope stand. It is better to 
have the scale numbered from one end to the other, to 
avoid the use of positive and negative quantities. A 
deflection is then taken by subtracting the reading of 



150 



ELECTRICAL MEASUREMENTS. 



rest from the reading in the deflected position, or con- 
versely. 

The north-seeking pole of the control magnet should 
be turned toward the north for greatest sensibility. If 
it is turned the other way it increases the strength of 
field at the needles, and so lessens the sensibility or the 
deflection for a given current. 

74. Calibration of any Galvanometer by Compar- 
ison with a Tangent Galvanometer/ — Connect a tan- 
gent galvanometer T, the galvanometer to be calibrated 
(r, a battery B^ and a suitable resistance i^, in series (Fig. 
72). Note the deflections of both T and G; vary the 

current by changing .R, 
and again read the de- 
flections. The resistances 
should be varied or so ad- 
justed that the deflections 
of (r may be as nearly 
as possible equidistant. 
Then if the constant of 
the tangent galvanometer 
has been determined pre- 
viously, the currents in 
amperes corresponding to the various deflections of Gr 
are known. Construct a plain elastic curve, with cur- 
rents as abscissas and deflections of G as ordinates. 
This will be the calibration curve of G-, from which may be 
read off the currents corresponding to other deflections. 

If the constant of T has not been determined, the 
calibration of Gr will be only relative and not absolute ; 




Fig. 72. 



Ayrton's Practical Electricity, p. 58. 



MEASUBEMENT OF CUBEENT. 



151 



that is, the deflections serve merely to compare currents, 
but not to measure them in amperes. 

It may happen that G- is more sensitive than T. In 
that case a suitable deflection of T produces too great a 
one in G. The difiiculty may be avoided by putting a 
shunt or by-path around (7, indicated at S. The calibra- 
tion Avill then be relative, unless the ratio of the resist- 
ances of G- and S is known. 



Example. 



G 


Tangent Gtalvanometeb. 












Currents (2). 


(2) -:- (1). 


Deflections (1). 


Deflections. 


Tangents. 






50 


2.2° 


0.038 


0.00192 


0.000384 


10 


11 


0.077 


0.00389 


0.000389 


15 


6 95 


0.122 


0.00616 


0.000410 


20 


9,8 


0.173 


0.00874 


0.000437 


25 


12.95 


0.230 


0.01161 


0.000464 


30 


16. 


0.287 


0.01451 


0.000483 


35 


19.3 


0.350 


0.01772 


0.000506 


40 


22.5 


0.414 


0.02096 


0.000524 



The curve (Fig. 73) expressing the relation between deflec- 
tions and currents is plotted as described above. 

75. Relative Calibration of a Galvanometer by 
Ohm's Law. — Connect a suitable constant potential 
battery to a slide-wire bridge PQ (Fig. 74), with sufii- 
cient resistance at R' to adjust the current through the 
bridge wire to a proper value. A key should be inserted 
in this circuit so as to keep the current flowing only 
so long as it is needed. Join the galvanometer to be 
calibrated and a resistance box to one end of the bridge 
wire at P, and the other end of tliis circuit to a suitable 
contact-maker on the wire. 

The experiment consists in placing the contact-maker 
A at successive equal divisions on the scale and observ- 



152 



ELECTRICAL MEASUREMENTS. 



ing the deflections of the galvanometer. A series of 
observations should first be made with the battery cur- 
rent flowing in one direction, and then another similar 
series with the current reversed. The mean of the read- 
ings should be taken for each division on the bridge scale. 























































































/ 










































/ 










































t 


/ 










































/ 










































/ 










































/ 










































/ 


/ 








































/ 


/ 










































/ 










































/ 










































/ 










































/ 










































/ 










































/ 










































/ 










































/ 










































/ 










































/ 








































/ 


/ 








































/ 


j^ 








































y 





















































1 











2 











3 











40 





Fig. 73. 

The differences of potential along the wire are, by- 
Ohm's law, proportional to the resistances passed over, 
or to the length of wire between the two points of the 
divided circuit. But the resistance in the circuit of the 
galvanometer remaining unchanged, the currents through 
it will be proportional to the P.D. between its terminals 
— that is, to the lengths of the bridge wire included be- 
tween the points of derivation A and P. 



MEASUREMENT OF CURBENT. 



153 



It is assumed that the E.M.F. of the battery remains 
constant, and that the resistance in circuit with it remains 
fixed. A storage battery is, therefore, to be preferred to 
a primary polarizable cell, and the student should care- 
fully guard against heating the conductor by keeping 



7£Z3^i 




Fig. 74. 



the circuit closed longer than is absolutely necessary. 
Since we have a divided circuit between A and P, an 
appreciable error will be introduced unless the resistance 
in circuit with the galvanometer is high in comparison 
with that of the bridge wire. 



Example. 

Calibration of a d''Arsonval Galvanometer. 



Readings 


Mean deflections 


Common 


Deflection 


on bridge wire. 


m mm. 


difference. 


per cm. 


10 cm. 


48.0 


48.0 


4.80 


20 


96.5 


48.5 


4.82 


30 


144.0 


47.5 


4.80 


40 


191.5 


47.5 


4.79 


50 


238.5 


47.0 


4.77 


60 


285.0 


46.5 


4.75 


70 


331.0 


46.0 


4.73 


80 


378.0 


47.0 


4.72 


90 


424.0 


46.0 


4.71 



154 



ELECTRICAL MEASUREMENTS. 



These observations are plotted with deflections of the gal- 
vanometer as ordinates and distances on the wire as abscissas 
(Fig. 75). The calibration curve is nearly straight, showing that 
the deflections are nearly proportional to the currents. 





















/ 


{ 


360 
320 

280 

240 
snn 
















/ 


/ 
















; 


/ 
















} 


/ 


















/ 


















/ 












160 

120 

80 

40 








/ 


/ 
















/ 


/ 
















/ 


/ 


















/ 


















/ 





















10 20 



40 50 

Fig. 75. 



80 90 100 



76. Calibration of a Galvanometer by Known 
Resistances. — The necessary apparatus consists of a 
battery of very low internal resistance, preferably a 
storage cell, and resistances reliably adjusted. The 
resistance of the galvanometer must also be known if it 
is enough to be appreciable in comparison with the 
remaining resistance in circuit. Connect the battery, 
the galvanometer, and the adjustable resistance in series. 
Adjust the resistance for successive readings of the gal- 



MEASUREMENT OF CURRENT. 



155 



vanometer and record galvanometer readings and total 
resistances in circuit. Then by Ohm's law the succes- 
sive currents are inversely proportional to the corre- 
sponding resistances ; and if the E.M.F. of the battery 
is known, the calibration will be in amperes. The inter- 
nal resistance of the battery is supposed to be negligible 
in comparison with the remaining resistance in circuit. 
The following data illustrate the method. The resist- 
ance of the instrument and connecting wires was found 
to be 1.6 ohms. This must be added to the resistances 
taken from the resistance box. 

£i2:ample. 



(a) 


(6) 


(c) 
Reciprocals of 




Readings of 


Total Resistance 


Resistance. 




Instrument. 


in Circuit. 


1 
b 




20 


860 +1.6 


.001160 




30 


563 


.001771 




40 


420 


.002372 




50 


334 


.002979 




60 


280 


.003551 




70 


238 


.004174 




80 


209 


.004748 




90.2 


185 


.005359 




100 


167 


.005931 




110.3 


151 


.006553 




119.8 


139 


.007112 




130 


128 


.007716 




140 


118.6 


.008319 




151 


110 


.008960 




161 


103 


.009560 




170.8 


97 


.010142 




180 


92 


.010684 




190 


87 


.011286 




199 


83 


.011820 





156 



ELECTRICAL ME A S UEEMENTS. 



Columns {a) and (c) have been plotted as coordinates (Fig. 
76), and the result is very accurately a straight line passing 
through the origin. The instrument of the table was a Weston 
milli-voltmeter, reading from 2 to 20 milli-volts, and the scale 
readings are directly proportional to the currents and therefore to 
the volts measured. 



200 
180 
160 
140 
120 
100 



40 



„,_1 — J — ^ — J — Ju, — L— -J \ 



10 



30 30 



50 60 ro 

Fig. 76. 



§0 100 110 



120 



77. Measurement of Current by Electrolysis. — 
When an electric current passes through a chemical 
compound in the liquid state, the compound is decom- 
posed. The process is called electrolysis^ and the com- 
ponent parts into which the substance is divided are 
called ions. These collect at the electrodes., or the con- 
ductors by which the current enters and leaves the 
electrolyte. 



MEASUBEMENT OF CURRENT. . 157 

The electrode by which the current enters is called 
the anode ; and the one b}^ which it leaves the electro- 
lyte is the cathode. 

Faraday demonstrated that the quantity of an ion de- 
posited is proportional to the quantity of electricity 
which has passed. Hence the quantity deposited in unit 
time is proportional to the current strength. 

He further showed that the same quantity of electricity 
deposits weights of different ions proportional to their 
chemical equivalents ; that is, proportional to the relative 
quantities which chemically replace one another. Thus 
the quantity which will release one gramme of hydrogen 
will deposit 32.5 grammes of zinc, 31.66 of copper, 108 
of silver, and so on. These quantities are the atomic 
weights of univalent substances and the half atomic 
weight of bivalent ones. It follows tliat if the weight 
of one of the substances deposited by one coulomb can 
be found by experiment, the known atomic weights of 
the chemical elements will give the electrochemical 
equivalents of the others, or the weights of the several 
elements which are released or deposited by one cou- 
lomb of electricity. 

When the electrochemical equivalent of some con- 
venient element has been ascertained, then the weight 
of it deposited in an observed interval of time serves as 
a measure of the quantity of electricity which has passed. 
If further the current has been maintained at a constant 
value, then this value may be determined by dividing 
the whole quantity of electricity by the time in seconds, 
or by dividing the weight of the ion by the product of 
the electrochemical equivalent and the time. The elec- 
trolytic process furnishes the practical method of deter- 
mining the international ampere (Art. 19). 



158 ELECTBICAL MEASUREMENTS. 

If w is the weight of tlie ion deposited, z its electro- 
chemical equivalent, and t the time of deposit, then the 
current will be 

~zi' 

78. The Silver Voltameter. — For currents as large 
as one ampere the cathode on which the silver is depos- 
ited should take the form of a platinum bowl not less 
than 10 cms. in diameter and from 4 to 5 cms. in depth. 

The anode should be a plate of pure silver some 30 
sq. cms. in area and 2 or 3 millimetres in thickness. 
This is supported horizontally in the liquid near the 
top of the solution by platinum wires passing through 
holes in the plate. To prevent the disintegrated silver 
or particles of silver oxide or carbon falling from the 
anode into the platinum bowl, the anode should be 
wrapped around with pure filter paper and secured at the 
back with sealing wax. 

The liquid should consist of a neutral solution of pure 
nitrate of silver, containing about 15 parts by weight of 
the nitrate to 85 parts of water. 

The resistance of the voltameter changes somewhat as 
the current passes. To prevent these changes having 
too great an effect on the current, some resistance 
besides that of the voltameter should be inserted in 
the circuit. The total metallic resistance of the circuit 
should not be less than 10 ohms. 

The method of making the measurement is as follows : 

The platinum bowi is washed with nitric acid and 
distilled water, dried by heat, and then left to cool in a 
desiccator. When thoroughly dry it is weighed care- 
fully. 



MEASTTBEMENT OF CURBENT. 159 

It is nearly filled with the solution and connected to 
the rest of the circuit by being placed on a clean copper 
support, to which a binding-screw is attached. The 
copper support must be insulated. 

The anode is then immersed in the solution, so as to 
be well covered b}^ it, and supported in that position ; 
the connections to the rest of the circuit are then made. 
Contact is made at the key, noting the time of contact. 
The current is allowed to pass for not less than half an 
hour, and the time at which contact is broken is ob- 
served. Care must be taken that the clock used is 
keeping correct time during the interval. 

The solution is now removed from the bowl and the 
deposit is washed with distilled water and left to soak 
for at least six hours. It is rinsed successively with dis- 
tilled w^ater and absolute alcohol, and dried in a hot-air 
bath at a temperature of about 160° C. After cooling in 
a desiccator the bowl is weighed again. The gain in 
weight gives the silver deposited. 

To find the current in amperes, this weight, expressed 
in grammes, must be divided by the number of seconds 
during wdiich the current has been passed and by 
0.001118. 

The result will be the time average of the current, if 
during the interval the current has varied. 

In determining by this method the constant of an in- 
strument the current should be kept as nearly constant 
as possible, and the readings of the instrument taken at 
frequent observed intervals of time. These observations 
should give a curve from which the reading correspond- 
ing to the mean current (time average of the current) 
can be found. The current, as calculated by the vol- 
tameter, corresponds to this reading. 



160 



ELECTRICAL ME A S UUEMENTS. 



Instead of dividing hj the time of deposit in seconds 
and by 0.001118, it is usually easier to divide by the time 
in hours (fractions) and by 4.025. 

Instead of the costly platinum bowl as cathode, a con- 
venient substitute, which is superior in some respects, is 
a flat silver plate, moiuited between two anode plates of 
pure silver, as shown in Fig. 77. The plates are mounted 

on a hard-rubber 
A ,.B strip A by means 

of stiff spring clips. 
By loosening the 
screw B^ the plates 
can all be removed 
together from the so- 
lution. The plates 
can be raised or low^- 
ered by means of 
a rack and pinion. 
This is a convenient 
method of effecting 
a fine adjustment 
of the resistance of 
the circuit in mak- 
ing and maintaining an electrical balance. The anode 
plates do not need to be covered with filter paper, since 
any dislodged particles will fall to the bottom of the jar. 
Great care is necessary in washing, drying, and Aveigh- 
ing the gain plate. It may be handled and weighed by 
means of a hook of stiff brass wire for suspension. This 
is a better plan than to run the risk of detaching parti- 
cles of silver by laying the plate down, except in the 
bottom of a glass tray in washing. This form of voltam- 
eter provides better insulation than those in which the 




Fig. 77. 



MEASUREMENT OF CUREENT. 161 

bowl rests on a base on which the nitrate of silver solu- 
tion is almost certain to be spilled by '.ack of extreme 
care. In this form neither the base n-oi- the standard 
forms any part of the conducting circuit. 

79. The Copper Voltameter. — When large currents 
are measured by electrolysis . the copper voltameter is 
employed instead of the corresponding one of silver, 
because the size of the plates required would make the 
latter too expensive. The copper voltameter scarcely 
equals the silver voltameter in accuracy, partly because 
of oxidation and partly because the electrochemical 
equivalent of copper is much smaller than that of silver, 
so that for a given current the quantity of copper depos- 
ited is less than that of silver, and it cannot be weighed 
with so small a percentage of error. On the other hand, 
the copper has the advantage of simplicity in manipu- 
lation. Silver is always deposited in a crystalline form, 
and requires careful washing and handling to avoid 
losses. It is difficult to make it adhere firmly to the 
gain plate or platinum bowl unless the surface is not less 
than 200 nor more than 400 sq. cms. per ampere. The 
deposited copper is much more firmly adherent, and 50 
sq. cms. per ampere will give good results. Thus for 
large currents, the copper plates need not be more than 
one-fifth as large as the silver. 

The solution is made by dissolving copper sulphate 
crystals in distilled water and adding one per cent 
of sulphuric acid. It may have a density varying 
from 1.1 to 1.2 without any difference in the nature 
of the deposit. A density of about 1.15 to 1.18 is to 
be preferred. 

The solution should not be used too often, since the 



162 ELECTHICAL MEASUREMENTS, 

acid is exhausted hj action on the plates ; and unless 
the solution is distinctly acid the results will be very 
irregular. i 

The loss plates should never have an area of less than 
40 sq. cms. per ampere. If they are smaller than this, 
the resistance of the cell becomes variable and the 
current camiot be kept constant. 

The gain plates, or cathode, should never be less than 
20 sq. cms. per ampere. An area of from 50 to 100 sq. 
cms. per ampere is best. The smaller the area the less 
firmly adherent is the crystalline copper deposit. When 
the deposit is continued for a long time the larger area 
should be used.^ At the current density of one-fiftieth 
of an ampere per sq. cm. there is a slight tendency for 
the deposit to thicken at the edges of the plates and 
become rough, but this tendency becomes less marked as 
the current density diminishes. A uniform and solid 
deposit is very desirable, and this is interfered with if 
the plates roughen at the edges. 

The plates may be prepared by rounding and smooth- 
ing the edges and corners, and then polishing thor- 
oughly with glass paper and washing in a rapid stream 
of water. They may then be rubbed with a clean 
cloth. On removing from the electrolytic cell, wash at 
once thoroughly in water containing a few drops of 
sulphuric acid, finally in distilled water, and dry on a 
clean blotting-pad. The plate may then be held before 
a fire and carefully warmed. It must not be weighed 
till it has cooled. 

For large currents a rectangular glass or earthenware 
vessel may be used to contain the solution, and the plates 

1 A. W. Meikle, The Electrolysis of Copper Sulphate, Physical Soc. of Glas- 
gow University. 



MEASUBEMENT OF CURBENT. 



163 



may be of the shape shown in Fig. 78 
in spring clips on one 
side, the anode and cath- 
ode plates alternating, 
one set connected by the 
clips on one side and the 
other set on the other. 
Each plate may then be 
lifted out and cleaned 
separately. The follow- 
ing table is given by Mr. 
Meikle, connecting the 
area, of the plate, the tem- 
perature, and the electro- 
chemical equivalent : 




Sq. cms. of cathode 


12° C. 


23° C. 


28° C. 


per ampere. 








50 


.0003288 


.0003286 


.0003286 


100 


.0003288 


.0003283 


.0003281 


150 


.0003287 


.0003280 


.0003278 


200 


.0003285 


.0003277 


.0003274 


250 


.0003283 


.0003275 


.0003268 


300 


.0003282 


.0003272 


.0003262 



The process of obtaining the current from the weight 
of copper deposited in an observed time is the same as 
in the case of silver. 

The following solution for a copper voltameter is said 
to give good results : ^ 

Copper sulphate 15 gms. 

Sulphuric acid 5 '* 

Alcohol 5 " 

Water 100 " 

'^Electrician (London), May 19, 1893. 



164 ELECTRICAL MEASUREMENTS. 

This can be used with a current density from 0.06 to 
1.5 amperes per square decimetre. 

80. To find the Constant or Reduction Factor of 
any Current Meter by Electrolysis. — If the currents 
to be measured by the instrument in question do not 
much exceed one ampere, the silver voltameter is to be 
preferred ; but for currents in excess of one ampere the 
copper voltameter ma}^ be used. 

When applied to a tangent galvanometer the operation 
consists in finding the reduction factor A^ which multi- 
plied by the tangent of the angle of deflection gives the 
current in amperes. With an electrod}Tiamometer the 
process has for its object the determination of the con- 
stant in the equation 

1= As/I), 

in which D is the torsion in divisions of the scale and A 
is the constant to be determined. When applied to a 
direct-reading ammeter it can find only the error of the 
scale corresponding to the number of amperes floAving 
through the voltameter. The apparatus may be set up 
as follows: 

5 is a storage battery of a sufficient number of cells 
to furnish the requisite current through the parallel 
resistances R and R' and the voltameter F(Fig. 79). 
When the E.M.F. of the battery and the approximate 
current which is to be measured by the voltameter are 
known the resistances JR, and R' can be determined 
beforehand. R' is put in parallel with R for the purpose 
of keeping the current constant through the voltameter 
and galvanometer. It may be either a carbon rheostat 
of the proper construction, or any other resistance 



MEASUREMENT OF CURRENT. 



165 



adjustable by insensible or at least very small gradations. 
Any small change in the current can thus be very readily 
compensated by adjusting the resistance B\ 
A convenient 



form for c u r - 
rents not exceed- 
ing three or four 
amperes may be 
made by wind- 
ing a flexible 
cable, such as 
heavy picture- 
wire, on an in- 
sulating tube 
supported by an iron rod through it and around insu- 
lating pins at the bottom (Fig. 80). The conductor is 







II 


© 


E 








fWY 






V -■ 


-\AAA 




Fig. 79. 



^ 




Fig. 80. 



thus wound non-inductively. If it were wound round 
and round on the frame or on a cylinder, it would pro- 



166 ELECTRICAL MEASUREMENTS. 

cluce a magnetic field within it. The long brass screw 
at the top is traversed by a contact-maker. Instead of 
a nut this co itains a screw pin, so that the contact-maker 
may slide readily from one end of the screw to the other 
by merely unscrewing the pin. When the pin is screwed 
in, the contact-maker may be moved slowly along the 
wires, so as to vary the portion in circuit, by turning the 
handle. 

If the constant of the electrodynamometer is to be 
determined, the instrument should be set up with the 
plane of its movable coil at right angles to the mag- 
netic meridian, or with its magnetic axis in the earth's 
magnetic meridian, and variable currents should be 
avoided. 

As a check, it is desirable to employ two electrolytic 
cells in series. One-half the weight of the electrolyte 
or metal deposited in the two is then taken for use in 
the formula with either the silver or the copper voltam- 
eter. 

Example I. 

To find the Reduction Factor of a Tangent GaIua?iomete?\ 

The galvanometer was set up in series with a silver vol- 
tameter, two Daniell cells, and a commutator for reversing the 
current through the galvanometer. The coil used was marked 
29.893 ohms. The current deposited silver for thirty minutes, 
and the deflections were read every minute, except when the 
current was reversed, when one reading was omitted. The 
observations were as follows : 



MEASUREMENT OF CUEEENT. 



167 





Deflections. 




Deflections. 


Time. 




Time. 




Left. 


Right. 


Left. 


Right. 


11.09 






25 


43.2 




10 


41.3 




26 






11 


41.3 




27 




44.0 


12 






28 




44.0 


13 




42 


29 




44.4 


14 




42 


30 




44.5 


15 




42.5 


31 




44.6 


16 




42.5 


32 




44.6 


17 




42.6 


33 






18 




42.7 


34 


44.2 




19 






35 


44.4 




20 


42.5 




36 


44.5 




21 


42.6 




37 


44.6 




22 


42.7 




38 


44.6 




23 


43.0 




39 


44.7 




24 


43.1 











Mean 43.34 43.37 

Mean deflection 43.36 

Tangent of mean deflection 0.94435 

Weight of cathode before deposit .... 30.3726 gms. 

Weight of cathode after deposit 30.4685 

Gain 0.0959 



Average current equals 



Therefore 



0.0959 
4.025 X h 

0.04765 



0.04765 = A tan 6. 



0.94435 



= 0.05046. 



Example II. 

To find the Constant of Siemens Electrodynamometer, No. 97 Q. 

Two copper voltameters were connected in series with the 
electrodynamometer, 14 cells of storage battery, and a resistance 
which served to regulate the current. 

The table gives the observations at one-minute intervals : 



168 



ELECTRICAL ME A S UREMENT8. 







Readings. 




Readings. 


v^Readings. 


^/Readings. 


81 


9 


79.8 


8.933 


81 


9 


79.5 


8.916 


80.5 


8.972 


79.5 


8.916 


80 


8.944 


79.5 


8.916 


80 


8.944 


81 


9 


80 


8.944 


81 


9 


80 


8.944 


81.5 


9.028 


80 


8.944 


81.5 


9.028 


80 


8.944 


81.5 


9.028 


80 


8.944. 


81.9 


9.050 


78.8 


8.877 


81.9 


9.050 


79.9 


8.939 


81.9 


9.050 


79.8 


8.933 


82.1 


9.061 


79.9 


8.939 


82.2 


9.066 


79.8 


8.933 


82 


9.055 



Mean 



Weight of cathode plate before deposit 
" after 
Gain 

0.955 



I. n. 

103.6476 83.4925 

104.6026 84.4475 

0.955 0.955 



I=AJI) 



Therefore 



h X 1.1838 
1.6134 



1.6134 amperes. 



8.977 



= 0.1797. 



81. Arrangement for Strong* or Weak Currents.^ — 
When a very strong or a very weak current is used, the 
apparatus illustrated in Fig. 81 may be employed. In 
the former case the current which it is desired to measure 
is larger than the capacity of the electrolytic cell ; in the 
latter case it is smaller than it is necessary to use for the 
purpose of obtaining an accurate result by electrolysis. 
The figure shows the arrangement for the first case of 
heavy currents, in which the current through the instru- 
ment for measuring current is nine times as great as 
through the two electrolytic cells in series. 

' Gray's Absolute Measurements in Electricity and Magnetism, Vol. II., 
Part II,, p. 428. 



MEASUREMENT OF CURRENT. 



169 



A set of parallel straight wires of German silver, plati- 
noid, or manganin are soldered to thick terminal bars of 
copper, 5, ^1 , h. , as shown, so that they can be connected 
in two groups in parallel. The wires in position must 
have accurately the same resistance. A sensitive reflect- 
ing galvanometer g of high resistance connects hi and ho . 
The resistances M and M' must be so adjusted that no 




r' 



£>2 G 



Fig. 81. 

current flows through g ; or, in other words, so that hi 
and ho are at the same potential. The current through 
(x will then be nine times the current measured by the 
electrolytic cells Fand V, or in the ratio of the con- 
ductances of the two groups of wires r and r^. 

G- is the galvanometer or other current measurer to be 
calibrated. 

82. Measurement of Current by Means of a 
Standard Cell. — A standard Clark cell will be de- 
scribed later (Art. 85). For the present, it is only 
necessary to say that a Carhart-Clark cell gives a con- 
stant E.M.F. of 1.440 volts at 15° C. (Latimer-Clark 
cell, 1.434 V.) Such a cell may be employed in connec- 



ITO 



ELECTRICAL MEASUREMENTS. 



tion with standard resistances to measure currents in 
amperes. 

The method consists in balancing the E.M.F. of a 
standard cell against the fall of potential over the whole 
or a part of a known resistance through which the cur- 
rent to be measured flows. 

Let r (Fig. 82) be the known resistance placed in the 
main circuit in which flows the current to be measured. 

This resistance should consist of a metallic conductor 
capable of carrying the current without undue heating. 




A/vVV^/vVV\^AAAAAVVV\AA/\^^^ << 



Fig. 82. 



If it is so mounted that it can be immersed in kerosene 
or oil the temperature can be kept nearly constant ; and, 
what is quite as important, it can be measured accurately. 
Two resistance boxes of high resistance are then 
placed in a derived circuit as a shunt to the resistance r. 
From the terminals of one, as Ri , another derived cir- 
cuit is set up containing a standard cell S and a sensitive 
galvanometer Gr. This circuit should also contain a 
key. The poles of the cell must be turned so that the 
P.D. over Ri shall be opposed to the E.M.F. of the cell- 
The balance is then made by adjusting R^ or R2 till no 



MEASUBEMENT OF CURRENT. 171 

current flows through the galvanometer on closing the 
key in its circuit. We have then 

E,: Bi + R, : : 1.44 : E, 
where E is the P.D. between C and D. 
Then ^=1.44^1+^'. 

If the temperature of the standard cell is not 15° C. a 
correction must in general be made. 
Finally, 

^_ 1^ R, + R. 
~ r ' R, ' 

It is evident that the resistance ?• must be such that 
the P.D. between its terminals shall be equal to or 
greater than the E.M.F. of the standard cell. 

Example. 

To determine the Consta?it of a Thomson ' ' Graded Galvanometer " 
{ammeter) without its Field-Magnet. 

AX B 



Formnla : I = 



Base number 



where A is the constant to be determined, I) the deflection, and 
by " base number" is meant the number indicating the position of 
the sliding magnetometer box on the base of the instrument. 

Data : i^i = 2110 ; R2 = 1254 ; and r = 10 ohms at 24° C. 

E.M.F. of standard cell at 20.5'^ C. = 1.437 volts. 

Therefore, I ='^ . 2110 + 1254 ^ ,^29 ampere. 
10 2110 ^ 

JD := 38.5 divisions. 

Base number = 32. 

Hence from the above formula, 

^^ 0-229 X 32 ^0.190. 
38.6 



172 



ELECTRICAL MEASUREMENTS. 



This constant is the value of the magnetic field at the needle 
when no current is flowins:. 



83. Second Method by Means of a Standard Cell. 
— This method, the connections for which are shown in 
the diagram (Fig. 83), admits of using a resistance r 
of such dimensions that the difference of potential 
between its terminals may be greater or less than the 




Fig. 83. 

E.M.F. of the standard cell or cells employed. The 
resistance must be capable of carrying the current during 
the time required to effect the balance without apprecia- 
ble heating ; or, better, it may be immersed in oil, with a 
stirrer, so that its tem})erature may be known. 

Set up two 10,000 ohm resistance boxes in series with 
a battery B^ of higher E.M.F. than the standard cell 
or the P.D. between A and B. From the terminals of 
it form a shunt circuit containing a sensitive high resist- 
ance galvanometer and a standard cell. It is better also 



MEASUBEMENT OF CUB BENT, 173 

to include a high resistance HR in this circuit. The 
poles of the standard cell must be turned in such direc- 
tion that the P.D. between the terminals of M opposes 
the E.M.F. of the cell. Then, keeping a total of 10,000 
ohms in the tivo boxes R and R'^ vary the part contained 
in each box till, on closing the key, the galvanometer G- 
shows no deflection. The P.D. between the terminals 
of R then equals the E.M.F. of the steaidard cell. Tlie 
high resistance HR may be so arranged, if necessary, 
that it can be short-circuited when a balance is nearly 
effected, so as to increase the sensibility. Then with the 
circuit closed through AB^ transfer the terminals of the 
derived circuit from ah to cd by means of the commuta- 
tor Q and balance again. The fall of potential over the 
resistance now in R will be equal to that over AB, 
But the two P.D.'s are proportional to the two resist- 
ances in R required to balance. Gall these Rx and R>. 

Then 

Rx :R>-. 1.44 : x, 

and a: — 1 .44 — = , 

Rx 

where x is the P.D. between A and B. Then as before 

r r ' R^' 

The E.M.F. of the standard cell must always be cor- 
rected for temperature. So also should the resistance r. 

This method is much more flexible than the first one, 
since it is not necessary to balance the E.M.F. of the 
cell directly against a part of the P.D. between the ter- 
minals of the resistance in the circuit in which the cur- 
rent to be measured is flowing. Hence witli the same 



174 ELECTRICAL MEASUREMENTS. 

resistance r a balance may be effected with a considera- 
ble range of current. This method may therefore be 
used to calibrate an ammeter Aju. 

Example. 

To test the Accuracy of a Weston Milli- ammeter. 
The ammeter was connected in series with r, a storage battery, 
and a resistance to control the current. 
Reading of milli-ammeter 0.828. 
r = 1.637 ohms at 25° C. 
Bi = 6885 ohms. 
i?2 =6502.5 ohms. 
E.M.F. of standard cell, 1.437 volts at 20° C. 

Hence / = l:43^Jo0^5 ^ ^ g^O ampere. 

1.637 V 6885 

84. Standard Resistances for the Preceding" 
Methods. — When large currents are measured by the 
preceding methods, special standard resistances adapted 
to carry the desired currents should be employed. Such 
standards have been designed at the Physikalisch-Tech- 
nische Reichsanstalt, in Berlin.^ They have a resistance 
of 0.01, 0.001, and 0.0001 ohm respectively, and are 
made of manganin in sheet form or cast. Special ter- 
minals, from the exact points between which the resistance 
is measured, are brought out to separate binding-posts for 
the measurement of the potential difference by compar- 
ison with a Clark cell. Any small E.M.F. of contact 
between the manganin and tlie copper terminals and 
leading-in conductors is thus left out of the comparison. 
The smallest resistance is adapted to carry a few thousand 
amperes. These standards are mounted in nickel-plated 

^ Elektrotechnische Zeitschrifty 1895. 



MEASUREMENT OF CURRENT. 



175 



cases (Fig. 84) which can be filled with oil. The large 
case for heavy currents is fitted with a cooling coil 




Fig. 84. 



through which water may be made to flow. It contains 
also a diminutive turbine-stirrer which can be driven by 
any small motor. 



176 ELECTBICAL MEASUBEMENT8. 



CHAPTER IV. 

MEASUREMENT OF ELECTROMOTIVE FORCE. 

85. The Clark Standard Cell. — In accordance with 
the decision of the Chamber of Delegates of the Chicago 
International Congress of Electricians (Appendix B), 
the Clark cell has become the legal standard of E.M.F. 
(Art. 19). The cell consists of zinc, or an amalgam of 
zinc with mercury, and of mercury in a neutral saturated 
solution of zinc sulphate and mercurous sulphate in 
water, prepared with both sulphates in excess. 

The preparation of the materials entering into the cell 
and the setting up of the standard will be described with 
some detail. 

A. Preparation of the Materials. 

1. The Mercury. — All mercury used in the cell 
should fost be chemically purified in the usual manner, 
and subsequently distilled in a vacuum. 

2. The Zinc. — Pure redistilled zinc-rods can be used 
without further treatment. For the preparation of the 
zinc amalgam add one part by weight of zinc to nine 
parts of mercury, and heat both in a porcelain dish luitil 
by gentle stirring at about 100° C. the zinc completely 
disappears in the mercury. 

3. The Mercurous Sulphate. — If the mercurous sul- 
phate, purchased as pure, is not colored yellow with a 
basic salt, mix with it a small 'quantity of pure mercury, 



2IEASUREMENT OF ELECTROMOTIVE FORCE. 177 

and wash the whole thoroughly with two parts by weight 
of cold distilled water to one part of the salt, by agitation 
or by stirring with a glass rod. Drain off the water and 
repeat the process at least twice, or until a very faint 
yellow tint appears. After the last washing drain off 
as much of the water as possible, but do not dry by 
heating. It is better to wash only so much of the salt as 
may be needed for immediate use. 

4. The Zinc Sulphate Solution. — Prepare a neutral 
saturated solution of chemically pure zinc sulphate, free 
from iron, by mixing in a flask distilled water with 
nearly twice its weight of pure zinc sulphate crystals, 
and adding pure zinc oxide in the proportion of about 
2 (fo by weight of the zinc sulphate crystals, to neutralize 
any free acid. The crystals should be dissolved by the 
aid of gentle heat, but the temperature of the solution 
must not be raised above 30° C. After warming for 
about two hours with frequent agitation, set the solution 
away over night. Then add mercurous sulphate, pre- 
pared as described in 3, in the proportion of about 12 % 
by weight of the zinc sulphate crystals, to neutralize 
any free zinc oxide remaining ; the solution should again 
be warmed, and should be filtered, while still warm, 
into a glass-stoppered bottle. Crystals should form as 
it cools. 

5. The Mercurous Sulphate and Zinc Sulphate Paste. 
— To three parts by weight of the washed mercurous sul- 
phate add one part of pure mercury. If the sulphate is 
dry it may be rubbed together with a mixture of the zinc 
sulphate crystals and concentrated solution of zinc sul- 
phate, so as to make a stiff paste, which shows through- 
out crystals of zinc sulphate and minute globules of 
mercury. If, on the contrary, the mercurous sulphate 



178 



ELECTRICAL ME A 8 UREMENT8. 



is moist, the paste should be made by adding the zinc 
sulphate crystals only, taking great care that they are 
present in excess and do not disappear after the paste 
has stood for some time. The mercury globules must 
also be plainly visible. The zinc sulphate crystals may 
with advantage be crushed fine before admixture with 
the mercury salt. 

The above process insures the formation of a saturated 
solution of the zinc and mercurous sulphates in water. 

B. To set up the Cell. 
The glass vessel containing the cell, represented in 
Fig. 85, consists of two limbs closed at the bottom and 

joined above to a common neck 
fitted with a ground-glass stop- 
per. The diameter of the 
limbs should be at least 2 cms., 
and their length 3 cms. The 
neck should be not less than 1.5 
cms. in diameter, and 2 cms. 
long. In the bottom of each 
limb a platinum wire of about 
0.4 mm. diameter is sealed 
through the glass. 

To set up the cell, place in one 
limb pure mercury, and in the 
other hot fluid amalgam contain- 
ing 90 parts mercur}^ and 10 parts zinc. The platinum 
Avires in the bottom must be completely covered by the 
mercury and the amalgam respectively. On the mer- 
cury place a layer 1 cm. thick of the zinc and mercurous 
sulphate paste described in 5. Both this paste and the 
zinc amalgam must then be covered with a layer of the 




Fig. 85. 



MEASUREMENT OF ELECTROMOTIVE FORCE. 179 

neutral zinc sulphate crystals 1 cm. thick; and the 
whole vessel must then be filled with the saturated zinc 
sulphate solution, so that the stopper, when inserted, 
shall just touch it, leaving, however, a small bubble to 
guard against breakage when the temperature rises. 

To prepare for placing the hot zinc amalgam in one 
limb of the glass vessel, after thoroughly cleaning and 
drjring the latter set it in a hot-water bath. Then pass 
through the neck of the vessel and down to the bottom 
a thin glass tube to serve for the reception of the amal- 
gam. This tube should be as large as the glass vessel 
will admit. It serves to protect the upper part of the 
cell from being soiled with the amalgam. 

To fill in the amalgam, a clean dropping-tube about 
10 cms. long and drawn out to a fine point has its fine 
end brought under the surface of the amalgam heated in 
a porcelain dish, and by pressing the rubber bulb some 
of the amalgam is drawn up into the tube. The point 
is then quickly cleaned of dross with filter paper, and is 
passed through the wider tube to the bottom and emptied 
by pressing the bulb. The point of the tube must be so 
fine that the amalgam will come out only on squeezing 
the bulb. This process is repeated till the limb con- 
tains the desired quantity of the amalgam. The vessel 
is then removed from the water bath ; and, after cooling, 
the amalgam must be fast to the glass, and must show a 
clean surface with metallic lustre. 

For insertion of the mercury a dropping-tube with a 
long stem will be found convenient. The paste may be 
poured in through a wide tube reaching nearly down to 
the mercury and having a funnel-shaped top. If it does 
not move doAvn freely it may be pushed down with a 
small glass rod. The paste and the amalgam are then 



180 ELECTRICAL MEASUREMENTS. 

both covered with the zinc sulphate crystals before the 
concentrated zinc sulphate solution is poured in. This 
should be added through a small funnel, so as to leave 
the neck of the vessel clean and dry. 

Before finally inserting the glass stopper it should be 
brushed round its upper edge with a strong alcoholic 
solution of shellac, and should then be firmly pressed in 
place. 

For convenience and security in handling, the cell 
thus set up may be mounted in a metal case which can 
be placed in a petroleum or paraffin oil bath. Its top 
may be provided with two insulated binding-posts to be 
connected with the two electrodes by the platinum wires, 
and the bottom should be perforated to allow the petro- 
leum or oil to enter freely. 

In order to ascertain the temperature of the cell, the 
metal case should enclose a thermometer which can be 
read from without. The thermometer may be fused 
into the glass stopper, or it may be entirel}^ separate 
with its bulb immersed in the petroleum or oil bath 
within the case. The latter method is to be preferred. 

In using the cell sudden variations of temperature 
should, as far as possible, be avoided, since the changes 
in electromotive force lag behind those of temperature. 

The E.M.F. of this cell is 1.434 volts at 15° C. 

For a small range of temperature above or below 
15° C. the following formula may be employed to reduce 
to 15°: 

E, = 1.434 [1 - 0.00080 (t - 15)] . 

Dr. Kahle gives for the formula connecting the E.M.F. 
at f with that at 15° the following : 

^, = ^- 116 X 10-"' (^ - 15) - 1 X 10-^ (^t-loy. 



MEASUREMENT OF ELECTROMOTIVE FORCE. 181 

This holds between 10° and 30° C. The E.M.F. of 
this cell decreases by about 0.00115 volt per degree C. 



86. The Carhart-Clark Standard Cell. — As a 
standard for practical commercial purposes a cell is 
needed which has the advantages of portability and a 
lower temperature coefficient than the normal Clark cell. 
These advantages have been secured in the following 
manner : 

A piece of No. 28 platinum wire is heated red hot in 
a blow-pipe flame, and is then sealed into the bottom of 
a small tube about 5 cms. long and 1.5 cms. in diameter. 
In contact with this is pure redistilled mercury. A 
layer about 1 cm. thick of pure neutral mercurous sul- 
phate mixed with neutral zinc 
sulphate saturated at 0° C. is 
placed on the mercury. The 
paste is then covered with 
purified asbestos ; on this rests 
the broad foot of the zinc, cast 
as shown in Fig. 86. To the 
top of the zinc is soldered a 
thin copper wire. For the 
purpose of holding the seal a 
cork disc surrounds the top 
of the zinc. This must be 
thoroughly boiled in distilled 
water to remove the tannin, 
and after drying may be satu- 
rated with pure paraffin. The zinc sulphate solution sur- 
rounding the zinc must be poured in through a small 
funnel before the zinc is inserted. Finally the cell is 
sealed by pouring in hot a cement composed of gutta- 




CastZn 



Fig. 86. 



182 ELECTBICAL MEASURE3IENTS. 

percha and Burgundy pitch, with enough balsam of fir 
added to make the compound flow when hot. After 
this has cooled, it is of advantage to add a mixture of 
finely powdered glass and sodium silicate. 

The temperature coefficient is reduced to one-half that 
of the Clark cell by the use of a zinc sulphate solution 
saturated at a temperature lower than any at which the 
cell is to be used. A convenient temperature for this 
solution is 0° C. In the normal Clark cell a rise of 
temperature causes more zinc sulphate to go into solu- 
tion. The consequent increase of density lowers the 
E.M.F. of the cell, and this effect is added to the real 
temperature coefficient which is due to the superposition 
of the two thermo-electromotive forces between the 
metal and the solution on the two sides of the cell.' 
Moreover the slowness with which the solution reaches 
the density corresponding with a new temperature causes 
the E.M.F. of the Clark cell to lag behind the tempera- 
ture change. Both of these difficulties are avoided by 
the employment of a solution saturated at zero degrees. 

The equation connecting the E.M.F. and temperature 
of the Carhart-Clark cell is 

^, = 1.440 j 1-0.000387(^-15) + 0.0000005(^-15)2 | . 

Near 15° C. a formula sufficiently accurate for practical 
purposes is 

U, = 1.440 j 1 - 0.0004 (t - 15) I . 

The temperature coefficient of this cell is thus one-half 
that of the normal Clark standard. 

iCarhart's Primary Batteries, p. 136; Amer. Jour, of Science, Vol. XL VI., 
p. 60. 



MEASUBEMENT OF ELECTROMOTIVE FORCE. 183 

87. A One-Volt Calomel Cell. — The calomel cell, 
consisting of mercury in contact with mercnrous 
chloride and zinc in zinc chloride solution, was invented 
by yon Helmholtz in 1882.' One of the present writers 
has inyestigated it with a yiew to adjust to exactly one 
yolt.' 

In 1879 D. H. Fitch patented a cell in which mer- 
curous chloride was used as the depolarizer, but in other 
respects it differed from the Helmholtz form. 

The E.M.F. of a chloride cell with zinc immersed in 
its chloride increases with decrease in density of the zinc 
chloride solution. Within limits, therefore, the E.M.F. 
of the calomel cell can be yaried by varying the density 
of the zinc chloride solution. An increase of about 
4.6 per cent in the density of the solution produces a 
decrease of 1 per cent in the E.M.F. The density 
required to give one yolt is 1.391 measured at 15° C. 

This cell is made in precisely the same form as the 
preceding. Such a cell is perfectly portable ; and cells 
in our possession over a year old show no appreciable 
change in E.M.F. compared with normal Clark cells. 

The temperature coefficient is small and is positive. 
The following equation connects the E.M.F with tem- 
perature for changes of a few degrees in the neighbor- 
hood of 15° C, or between 10° and 30° C. : 

^=1 + 0.000091 (^-15). 

A near approach to the coefficient is 0.01 per cent per 
degree. A neglected variation of 10 degrees can cause 
an error of only 0.1 per cent. 

Since the modified Clark cell described in the last 

iSitzber. cler ATcad. der Wiss., p. 26, Berlin, 1882. 
2 Amer. Jour, of Science, Vol. XLVI., p. 60. 



184 ELECTRICAL MEASUREMENTS. 

article has a negative coefficient and the calomel cell a 
small positive one, it becomes possible to combine the 
two varieties in such a way that the combined set shall 
have a zero coefficient. Let x equal the number of calo- 
mel cells required to offset one Carhart-Clark. 

Then 0.000094 x = 1.44 x 0.00039, 

or x = Q nearly. 

88. The Weston Standard Cell. — Mr. Edward 
Weston has invented a standard cell consisting of mer- 
cury in contact with mercurous sulphate and cadmium 
amalgam immersed in a saturated solution of cadmium 
sulphate. The H form of the cell, similar to Fig. 85, has 
been selected as the best. A platinum wire is sealed into 
the bottom of each limb. In one limb is the pure mercury, 
and resting on it the mercurous sulphate paste mixed 
Avith the cadmium sulphate solution. In the other limb 
is the cadmium amalgam. The vessel is finally filled so 
as to connect the two limbs with the cadmium sulphate 
solution, and is sealed in the usual manner. The only 
difference in the structure between this cell and the 
Clark is that cadmium and cadmium sulphate are used 
in place of zinc and zinc sulphate. The scheme of the 
cell is as follows : 

-\^Cd- CdSO, - Hg,SO, -H;i\+. 

Weston's patent gives the E.M.F. of the cell as 1.019, 
and the temperature coefficient 0.01 per cent per degree 
centigrade. 

This cell has also been investigated by Jager and 
Wachsmuth ^ at the Berlin Reichsanstalt. An amalgam 

1 Zeit.filr Instrumentenhunde, November, 1894. 



3IEASUBEMENT OF ELECTROMOTIVE FORCE. 185 

of 1 part of cadiuiiim to 6 parts of mercuiy was covered 
with a layer of cadmium sulphate crystals. The mer- 
curous sulphate was rubbed together with cadmium 
sulphate crj^stals, metallic mercury, and concentrated 
cadmium sulphate solution, so as to form a stiff paste. 
This was placed on the mercury of the positive pole. 
The remainder of the H element was filled with con- 
centrated cadmium sulphate solution, the negative pole 
containing the cadmium amalgam. 

Between 0° and 26° the temperature coefficient is 
expressed by the following formula : 

U,= Uo - l''2b xlO-'t- 0.065 X 10 ~' f. 

Near 20° the change of E.M.F. per degree C. is only 
about 0.00004 volt. The following table shows the com- 
parative temperature coefficients of the Clark and the 
Weston cell in xoVo P®^ cent: 





Temperature CoEFnciENX. 


t 


Clark. 


Weeton. 


0" 
10* 
20° 
30" 


— 70.9 

— 77.9 

— 84.9 

— 91.9 


— 1.3 

— 2.5 

— 3.7 

— 5.0 



Near 20° the E.M.F. of the cadmium element changes 
only about 2V as much as the Clark element for the same 
temperature variation. When two per cent of zinc was 
added to the cadmium the increase of E.M.F. was only 
about 0.0004 volt. The cadmium sulphate of commerce 
contains only small traces of foreign substances, and 
these produce no appreciable effect on the E.M.F. It 



186 ELECTRICAL MEASUREMENTS. 

is very essential, however, that the cadmium sulphate 
solution should be thoroughly neutral. Any trace of 
acid raises the E.M.F. To neutralize any acid cadmium 
hydroxide is used, and the filtered solution is treated 
with mercurous sulphate for the reduction of any basic 
salt that may have been formed. When the salt is thus 
treated different cells agree to within 0.0001 volt. 

The solubility of cadmium sulphate changes only 
slightly with temperature. This is one reason for the 
smallness of the temperature coefficient, and in con- 
sequence the cell quickly reaches an electrical equilib- 
rium after a variation of the temperature. 

The constancy of the Weston cell can only be deter- 
mined after long trial. Observations extending over 
four months showed that the element remained constant 
within 0.0001 volt. Compared with the Clark element 
its E.M.F. was found to be 1.022 volts. 

89. Comparison of E.M.F. 's by a Galvanometer 
in Shunt. — Let there be two or more cells the E.M.F. 's 
of which are to be compared. Connect one of them in 
series with a 4^esistance of from 10,000 to 15,000 ohms 
and another small resistance R (Fig. 87). It is not 
necessary to know the value of either of these resistances, 
but one of them should be large enough to prevent 
appreciable polarization of the cells during the time 
required to take a reading with the circuit closed. A 
d'Arsonval galvanometer, or some other, aperiodic form, 
is connected in a circuit joined as a shunt to the small 
resistance R. 

Close the key K and observe the deflection di. This 
should not exceed about 200 scale parts, with the scale 
one and a half metres from the mirror. It is best to 



MEASUREMENT OF ELECTROMOTIVE FORCE. 187 

take a series of observations for di and to make use of 
the mean. Next replace B with another cell and repeat 
observations for do. Then 

E,:E,:id^:d,. 

This method neglects any. difference in the internal 
resistance of the cells. If this resistance is small no 
appreciable error will result. But if the battery itself, 
or one of the cells compared, should have a high internal 




Fig. 87. 

resistance the method cannot be used. A comparison of 
a Daniell cell, for example, with a standard Clark, 
having an internal resistance of 2000 ohms or more, 
would give a result which would make the E.M.F.'s of 
the two cells apparently more nearly equal than they 
really are. But so long as the internal resistance of the 
cells compared is negligible in comparison with the other 
resistance in circuit, then no change in the circuit is 
made in substituting one cell for another except a change 
in the E.M.F. ,* and if the currents are proportional to 



188 



ELECTRICAL MEASUREMENTS. 



deflections, the E.M.F.'s, being proportional to the cur- 
rents, are also proportional to the corresponding deflec- 
tions. 

Example. 
R=20 ohms : R' = 15,000 ohms. 



Cell. 


Deflection. 


E.M.F. 


Daniell, 


64 


1.1 volts. 


" Diamond '' Carbon, 


67 


1.15 " 


Gassner Dry Cell, 


75 


1.29 " 


Ajax Dry Cell, 


63 


1.08 " 



The Daniell cell was freshly set up, but the others were old 
cells. 

90. The Condenser Method of comparing" E.M.F.'s. 

— Let (x be a sensitive galvanometer with a small damp- 
ing coefficient. Connect with the condenser and the 
battery B by means of a charge and discharge key 

^(Fig. 88). The con- 
denser will need to have 
a capacity of from 0.05 
to 0.3 of a microfarad. 
Observe the first swing 
several times Avhen the 
condenser is discharged 
through the galvanom- 
eter and take the mean 
for d^. The complete pe- 
riod of swing of the gal- 
vanometer, for convenience in reading, should be from 
5 to 10 seconds. Next repeat the observations with a 
second battery and let the mean of the deflections be c?2 • 
Then if Ei and E. are the E.M.F.'s of the two cells, 




MEASUBEMENT OF ELECTJiOMOTITE FORCE. 189 

To save time in waiting for the galvanometer needle 
to come to rest after each observation, a small coil may 
be placed near the needle, and a single cell may be con- 
nected in circuit with it. By tapping the key in this 
control circuit at the proper moment the needle may be 
quickly brought to rest. 

If the ballistic form of the d'Arsonval galvanometer 
be used, the motion of the coil may be arrested by short- 
circuiting the galvanometer by means of an extra key 
for the purpose. 

In this method the first swing of the needle from rest 
is nearly proportional to the quantity of electricity dis- 
charged through the galvanometer; and, since the capacity 
of the condenser remains unchanged, the quantities are 
proportional to the E.M.F.'s charging the condenser. If 
instead of a change in electromotive force another con- 
denser of different capacity be used, the deflections d^ 
and do will be proportional to the capacities of the two 

condensers. 

Example I. 



Cell. 

Clark, 
"Diamond' 


' carbon, 


Deflection. 

120 mm. 
114.5 mm. 


'herefore 120 : 


: 114. 


.5 :: 1.434: x ( = 


= 1.368 volts). 


CeU. 
Clark, 
Daniell, 




Example II. 

Deflection. 
265 

205 


E.M.F. 

1.428 (at 20° C.) 
1.105 



91. Lord Rayleig-h's Potentiometer Method. — The 
preceding methods are deflection methods and do not 
admit of great accuracy. If the deflection is 200 scale 
parts, and if it can be read to only a single division, 
then no greater accuracy than one-half per cent can be 



190 



ELECTRICAL MEASUREMENTS. 



secured. Zero methods are much to be preferred, and 
the following one leaves nothing to be desired, where 
the E.M.F.'s to be compared are only a few volts. Let 
R and R' (Fig. 89) be two well-adjusted resistance 
boxes of 10,000 ohms each. Connect them in series 
with a cell having a higher E.M.F. than either of the 
E.M.F.'s to be compared. A total resistance of 10,000 
ohms must be kept in circuit. A shunt circuit is taken 
from the terminals of one box Z?, and in this is placed a 
sensitive galvanometer, a key, one of the cells to be 




^ 







K *- 



F'g. 89. 

compa.red, and usually a high resistance to protect the 
cell from polarization, if a standard, as well as to avoid 
too large a deflection of the galvanometer. The cell 
B^ should be so connected that its E.M.F. may be bal- 
anced against the P.D. between the terminals of R. 
Obtain a balance, so that the galvanometer shows no 
deflection on closing the key K^ by transferring resist- 
ance from one box to the other, being careful to keeiJ the 
sum of the tivo 10,000 ohms. When a balance has been 
secured to the nearest ohm, the E.M.F. of the cell B^ 
equals the fall of potential over the resistance in R. 



IIEASUBEMEXT OF ELECTROMOTIVE FORCE. 191 

Repeat the operation with a second cell or other source 
of E.M.F. Then if Ri and Ro are the resistances in R 
in the two cases to balance, we have 

The resistance in the circuit is kept so large that no 
appreciable polarization takes place while the comparisons 
are being made. Then the P.D. between the terminals 
of R is strictly proportional to that portion of the 10,000 
ohms contained in the box R. If the galvanometer is sen- 
sitive to a change of a single ohm from R to i?', or the 
reverse, then the E.M.F. of the battery in the main cir- 
cuit should be only slightly higher than that of the 
highest E.M.F. to be compared. Larger numbers will 
•then be obtained to represent the E.M.F.'s, and hence 
greater accuracy in the result. 

If one of the cells compared is a standard with known 
E.M.F., the method gives the E.M.F. of each of the 
cells compared. Two cells to be compared may be con- 
nected in opposition to each other. In this way the 
difference of E.M.F. between them may be compared 
with the E.M.F. of either. 



Cell. 

No. 30 Clark, 
No. 3 Calomel, 


Examples. 

Temp. C. 
15« 

150 


Res. to balance. 

9475 
6607 


nee 


9475 : 6607 : : 1.434 : x, 




x = 0.9999 volt. 




Cell. 

No. 30 Clark, 
No. 7 Calomel, 
No. 9 " 
No. 10 
No. 11 


Temp. C. 
17.7« 
190 


Ree. to balance. 

9151 
6395 
6396 
6396 
6395 



192 



ELECTRICAL MEASUREMENTS. 



i; (Clark) = 1.434 [1 — 0.00077 (17.7 — 15)] = 1.431. 

Hence 9151 : 6396 : : 1.431 : x, 

or X = 1.0002 volts at 19° C. 

for Nos. 9 aud 10. 

And 9151 : 6395 : : 1.431 : x, 

or a; = 1.0000 volt 

for Nos. 7 and 11 at 19« C. 

92. Comparison of B.M.F.'s by Rapid Charge and 
Discharge. — ■ Two platinum wires dip into mercury cups 
a and h (Fig. 90) . The wires are attached to the prongs of 

a large tuning-fork, 




Fig. 90. 



and are insulated 
from them. When 
the prongs separate, 
one of the wires dips 
into the cup h and 
completes the con- 
nections so as to 
charge the condenser 
C. As soon as the 
prongs approach 
each other, connection is broken at h and the other wire 
enters the cup «, thus discharging the condenser through 
the galvanometer. If this operation is repeated a suffi- 
cient number of times a second, a steady deflection of 
the galvanometer will result. Let the deflection with a 
standard cell be <ii, and let E^ equal 1.44 volts. Re- 
place the standard with the cell to be compared, and 
obtain the deflection again and let it be d^. 
Then if x be the E.M.F. of the cell, 
d^'. d.i'' 1-44 : a-. 



or 



1.44^ 



IIEASUEEMENT OF ELECTBOMOTIVE FORCE. 193 

Great care must be taken to so adjust the contacts 
that one platinum wire will leave the mercury surface in 
b before the other touches the mercury surface in a, 
otherwise the E.M.F. of the cell would be applied 
directly to the galvanometer. The accuracy of the 
method is dependent upon keeping constant the num- 
ber of charges and discharges per second, since with a 
fixed capacity and E. M. F. the quantity discharged 
through the galvanometer in one second is proportional 
to the number of times the condenser is discharged. 

Example. 

Cell. Steady Deflectioa. E.M.F. 

Carhart-Clark, 350 1.44 volts. 

AjaxDiy, 310 1.27 " 

Bichromate, 430 1.77 " 

" Diamond " Carbon, 295 1.21 " 

Leclanche, 380 1.56 " 

93. Measurement of B.M.P. of a Standard Cell by 
a Kelvin Balance. — The apparatus at the bottom of 
Fig. 91 is set up as in Lord Rayleigh's metliod of com- 
paring E.M.F.'s. Find first with key ^open the num- 
ber of ohms in the box B required to balance the E.M.F. 
of the standard cell S in the shunt circuit. Then close 
key K and balance again while the current is flowing 
through the centi-ampere balance TB and the standard 
coil immersed in oil. The connections are made in 
the figure on the assumption that the fall of potential 
between the terminals of the coil O is less than the 
E.M.F. of the standard cell. Then when a balance is 
secured, the E.M.F. of the standard cell is balanced 
against the P.D. between the terminals of the coil 
plus the P.D. between the terminals of B. At the same 



194 



ELECTRICAL MEASUBEMENTS, 



time that this last balance is made, the current is meas- 
ured by means of the centi-ampere balance. 

A high resistance should be put in circuit with the 
galvanometer and standard cell, but it can always be cut 
out when the balance is nearly complete. 



II 1 


MM 



^a 




Fig. 91. 



Then if R is the resistance of coil C, R^ and Ro the 
resistances in B to balance with key K open and closed 
respectively, and I the current measured by the centi- 
ampere balance, w^e have 



E=IR 



Ri 



R,-Ro 



IR is the P.D. in volts betw^een the terminals of the 
coil C. This is represented by the loss of potential over 



MEASrBEMENT OF ELECTROMOTIVE FORCE. 195 

the resistance (^j — i^.,)- But the E.M.F. of the stand- 
ard equals the fall of potential over the resistance Ri in 
the auxiliary circuit of the Rayleigh method. Hence 

TO 

the P.D., IR., must be multiplied by the fraction — — ^--— 

Ri — R.2 

to obtain the E.M.F. of the standard. 

The operations ma}- be performed in a slightly differ- 
ent way. First, balance in the auxiliary circuit with the 
standard cell alone, as in the other case. Next, cut out 
the standard cell entirely, close key K and balance again. 
The current through the Thomson balance must then be 
reversed as compared with the figure. Let R^ and R^ be 
the resistances in the auxiliary circuit to balance in the 
two cases. Then 

E^IR^. 
R2 

The accuracy of this method can be no greater than 
that of the centi-ampere balance, even with resistances A 
and B accurately adjusted. 

The reverse reasoning gives a test of the accuracy of 
the balance. Given the E.M.F. of the standard cell, the 
equation determines the current. 

Example. 

Standard Cell, No. 25. 

Resistance in B to balance with K open 9416 

Resistance in B to balance with K closed 1802 

Temp, of standard cell 17.2° C. 

Temp, of coil C 17.2° C. 

Coil C equalled 10 ohms at 9° C. 

Temperature coefficient, 0.0002. 

Hence at 17.2° the resistance of the coil was 10.0164 ohms. 

Current through centi-ampere balance, 0.1162 ampere. 



196 ELECTRICAL MEASUREMENTS. 

Hence the electromotive force of the cell was 

0.1162 X 10.0164 X ?^1? = 1.4393. 

9416 — 1802 

This is at 17.2° C. At lo^ C, 

E = 1.4393 [1 4- .00039 (17.2 — 15)] = 1.4405 volts. 

94. Measurement of the B.M.F. of a Standard 
Cell by Means of the Silver Voltameter. — This 
method of measuring E.M.F. consists in comparing the 
P.D. between the terminals of a known resistance with 
the E.M.F. to be measured. To get the P.D. we must 
know not only the resistance between the two points, 
but the current flowing. The current is measured by- 
means of the silver voltameter, while the intermediate 
means of comparing the P.D. with tlie E.M.F. of the 
cell is the Rayleigh method of comparing E.M.F.'s, as in 
the last method. 

First, there must be provided as constant an E.M.F. 
as possible, so that the current to be measured by the 
voltameters may be nearly constant. Let Bi (Fig. 92) 
be a storage battery of a number of cells connected in 
series with a resistance R' and the standard or accu- 
rately known resistance R. It is desirable to include in 
this circuit also a carbon resistance, or some other one 
capable of changing continuously, or at least by very 
small steps. Fl and V-2 are the silver voltameters. B}" 
means of the commutator either a resistance r or the two 
voltameters can be thrown into circuit. By this means 
the current can be adjusted to the desired value before 
the voltameters are put into circuit. The resistance r 
should be made, as nearly as convenient, equal to that 
of the two voltameters. The advantage in using a num- 
ber of storage cells and a considerable resistance R' is 



MEASUREMENT OF ELECTROMOTIVE FORCE. 197 



that any small change in the resistance of the voltam- 
eters, or any small difference between their resistance 
and r, will be nearly or qnite inappreciable. The other 
part of the apparatus consists of the two 10,000-ohm 
boxes, A and B^ 
Avith one or two 
cells of Leclanche 
battery, a sensi- 
tive galvanometer 
(t^, a standard cell 
S, the E.M.F. of 
which is to be 
measured, and a 
commutator as 
shown, made by 
boring holes in a 
block of parafhn. 
By connecting ac 
and 5c?, the E.M.F. 
of the standard 
cell is first bal- 
anced against the 
difference of po- 
tential between 
the terminals of 

the box A. At the same time the temperature of the 
cell is noted. Then by connecting a and h to e and /, a 
balance can be made between the fall of potential over 
the resistance R and over that in A. When the prelim- 
inary balance has been secured and the temperature of 
R taken, the connections may be made with the voltame- 
ters and the current sent through them. The balance 
for the P.D. of R is again obtained. If the change of a 




Fig. 92 



198 



ELECTEICAL 3IEA S UREMENTS. 



single ohm in A reverses the deflection of the gal- 
vanometer, the exact balance may be effected by means 
of the carbon resistance mentioned above. The current 
should be allowed to flow for half an hour, and it may 
either be kept constant by means of the adjustable resist- 
ance, or it may be observed at equal time-intervals by 
means of the resistance in A required to balance. The 
balance for S should be tested occasionally, and the 
temperature of the cell should be kept constant if 
possible. 



aToo 



























■^jf.^ 


^ - 












— X — — 


e — 







-Q. 








"^ 




i— <^ 




















^^ 



I lir. 50 



55 



gnr. 



10 15 

Fig. 93. 



5750 



25 



30 



The resistance K should be made of manganin wire 
immersed in paraffin oil, and the case should be pro- 
vided with a stirrer to equalize the temperature. Any 
small change in this resistance is practically negligible, 
but allowance may be made for it, since the temperature 
coefficient of the manganin wire is supposed to be known. 

Fig. 93 shows the method of plotting the observations 
for a normal Clark cell and for the current. The mean 
value for the Clark is 5751.5 and for the current 3691.2. 
These values represent the mean ordinates for the two 
curves. 



MEASUREMENT OF ELECTROMOTIVE FORCE. 199 

Let Bi be the resistance in box A required to balance 
the Clark cell, and M, the resistance required to balance 
jRI of the known resistance H. 

Let M be the mass of silver deposited, t the time of 
deposit, and z the electrochemical equivalent of silver in 
grammes per coulomb. Then 

z = 0.001118. 
M= Itz. 



Therefore 
and 



R, zt 



The value of E thus found requires correction to 
reduce to temperature 15° C. 

Examples.! 

In the experiment to which the two curves of Fig. 93 relate R^ 

= 5751.5; i?2= 3691.2; i^ = 0.9877 at 17" C. 

Temp, of Clark, 16.45° C. 

Jf= 2.8095 gms. 

^ = 2700 seconds. 

Hence 

^ = 0.9877 5I^M. ^-^Q^^ =1.4324. 

3691.2 2700 X 0.001118 

Correction to 15° C. with coefficient 0.00077 = 0.0016. 
Hence E = 1.4324 -4- 0.0016 = 1.4340 volts at 15° C. 



Again, Ri = bl22.b; i?2 = 3904.5. 
i? = 0.9877 at 17° C. 
Temp, of Clark, 16.5° C. 
M= 2.6071 and ^=2357 seconds. 



^Glazebrook and Skinner, Phil. Trans., Vol. 183 (1892) A, pp. 567-628. 



200 



ELECTRICAL MEASUREMENTS. 



Then ^ = 0.9877 



5722.5 



2.6071 



1.4322. 



3904.5 2357 X 0.001118 
Correction to 15° C. = 0.0017. 
Whence E = 1.4322 -f 0.0017 = 1.4339 volts. 

95. Electrostatic Voltmeters. — The forces operat- 
ing in an electrostatic voltmeter are due to the attrac- 
tion and repulsion between static charges. Like the 



/ 



/' 







1 


11 




' M' 






■MiMiBfe^/ 


II: Ml^H^HMAM 








\ 






























\ 






\ 








^^'■'■■■■■'■''1 


1 ■"■-■■■■-■ a E^2 


1 


1,. 


L H/n? 


j 




-' 


>r 


J 


\ 1 


/ 


i 


3 


^S 




L^ 




\ r- 


rlk 


^j/^ 


1 1 


F— |y M 




i 


L_ S/ I'l 


/^ — J 


I 






B 7 ; 














i 










' 






H 










/ ' 






xT-\ 




/ ' 




,ll 


U 


— n 

y 


III / . ._.; 



Fig. 94. 



electrodjnamometer, it is applicable to both direct and 
alternating currents. It has no self-induction and takes 
no appreciable current, CA^en on an alternating current 
circuit, because of its small capacity (Art. 111). 

The instruments illustrated in Figs. 94 and 96 may 
very properly be called electrostatic electrodynamome- 
ters. Each contains a mirror from Avhich a beam of 
light from a lamp is reflected to a fixed scale ; and in 
using them the spot of light is brought back to the zero 



3IEASUBEMENT OF ELECTROMOTIVE FORCE. 201 

or initial position by turning the torsion head before the 
reading is taken. The beam of light, about a metre 
long, takes the place of the pointer of a Siemens dyna- 
mometer. 

Referring to Fig. 94, which consists of a horizontal 
and a vertical section, it will be seen that the fixed por- 
tions of the electrostatic part of the instrument consist 
of four half-circular flat boxes, three inches in diameter 
and half an inch in depth inside. The lower pair is 
supported on ebonite pillars, and the upper one is car- 
ried on the lower by means of lead-glass rods set into 
appropriate sockets. 

The needle consists of two half-circles of very thin 
aluminium mounted on a wire of the same metal, as 
shown in the lower left-hand corner of the figure. It is 
evident that when the half-circular boxes are cross- 
connected and one pair of these inductors is electrically 
connected with the needle, the forces acting on the 
movable system all tend to turn it in one direction. 

The needle is suspended by a phosphor-bronze wire, 
about 0.038 mm. in diameter, from a brass torsion-head 
with a hard-rubber top. The suspending wire is per- 
fectly free except at the point of support at the top of 
the brass head. The axis of the needle is connected 
below by means of a platinum-silver spiral to the cup 
containing paraffin oil as a damper. The damper itself 
is a horizontal disk supported by two wires from the 
axis of the needle, and having at its centre a hole 
through which passes the pin holding the lower end of 
the spiral. The needle is charged through this spiral; 
and, since the instrument is a zero one, the spiral does 
not affect its sensitiveness if the beam of light compos- 
ing the pointer can be brought accurately back to zero 



202 ELECTRICAL MEASUREMENTS. 

before the reading is taken ; for the instrument is set up 
so that the spiral is entirely without torsion when the 
beam of light is at the zero of the scale. The torsion 
scale rests on the hard-rubber top and is divided into 
400 equal divisions. The pointer is set to the zero of 
this scale after all other adjustments have been made. 
A key, shown in the charging position, is made to dis- 
charge the semi-circular inductors by turning it through 
180°. 

When the instrument is charged, the system swings, 
twisting both the supporting wire and the steadying 
spiral at the bottom. This spiral has more torsion than 
the wire. The torsion head is turned till the spot of 
light returns to zero, and the twist of the suspending 
wire is then read by the pointer on the circular scale. 
The spiral is without torsion when the torsion head 
stands at zero, but it serves to overcome the surface 
viscosity of the damping fluid, and to give a constant 
zero reading. The instrument is practically dead-beat 
and its performance is very satisfactory. The one rep- 
resented in Fig. 94 was intended to measure up to 1,100 
volts. Fig. 95 is its calibration curve. Since the in- 
strument is used idiostatically, this curve, like that of 
the electrodynamometer, should be a parabola. It de- 
parts from a parabola only very slightly. The constant 
increases a little on the upper readings. The points on 
the upper part of the curve were obtained by means of 
a platinoid resistance of 4,000 ohms, wound non-induc- 
tively on three frames supported in a horizontal position, 
so that all portions of the wire remain at the same tem- 
perature. This wire is divided into four sections, and 
the resistance of each section is accurately known. The 
smallest is about y^ of the entire amount. The whole 



MEASUREMENT OF ELECTROMOTIVE FORCE. 203 

was connected across the mains leading to an alternating 
dynamo, while conductors led from the terminals of the 
smallest section to a Kelvin multicellular voltmeter. 
The performance of this particular multicellular instru- 
ment is not satisfactory, partly because of an uncertain 
zero. Hence the vagaries of the points on the upper 
part of the curve. The points nearer the origin were 
taken by comparison with a Weston voltmeter and with 



1200 






















1000 


















^.--» 


^^ 


fiOO 












^ 


^ 




•-^^ 




VC 

600 


LIS 






^ 


^ 












'100 




y" 


^ 
















?on 


/ 


/r 




















/ 





















^0 



400 GOO 

TWIST 



Fig. 95. 



1000 



additional known resistance in circuit with it. A later 
calibration by means of the smaller instrument (Fig. 
96) gave a better result. 

Fig. 96 represents a similar instrument of smaller 
dimensions designed to measure from about 20 to 100 
volts. Its principle is identical with that of the other, 
and its construction is similar. The suspending fibre is 
in- this case quartz. Instead of semi-circular boxes for 
the inductors, parallel semi-circular plates are secured at 
fixed distances, and the entire system of inductors is 



204 



ELECTRICAL MEASUREMENTS. 



hung from the hard-rubber cross-bar which is adjustable 

on the supporting 
brass pillars carrying 
the top plate, scale, 
and torsion head. Fig. 
97 is its calibration 
curve. The suspend- 
ing fibre has since been 
replaced hj a slightly 
thicker one, so that 
one revolution of the 
torsion head corre- 
sjDonds almost exactly 
to 100 volts. Vertical 
cylindrical quadrants 
'^'^' ^^" and a vertical cylin- 

drical needle were first tried,^ but these did not prove 




80 
60 

V( 

40 
20 
















I 














^ 


^ 


LTS 




^ 




^^ 








/ 


^ 














/ 














1 



«« TWIST *" 

Fig. 97. 



800 



400 



SO satisfactory as the horizontal form of inductor plates 
and needle. 



Proceedings of the International Electrical Congress, 1893, p. 208. 



MEASUEE2IENT OF ELECTBOMOTIYE FOBCE. 205 

96. Calibration of a Voltmeter by Means of Stand- 
ard Cells. — The method consists in balancing the elec- 
tromotive force of one or more standard cells against a 
fraction of the potential differences applied to the bind- 
ing-posts of the voltmeter, and determining this fraction 
by means of well-adjusted resistance boxes. Let M and 
B' (Fig. 98) be two good resistance boxes, the first pref- 
erably as large as 100,000 
ohms. The range of the 
second one will depend 
upon the range of the cali- 
bration and the number of 
standard cells used. B is 
a storage battery of a suffi- 
cient number of cells to 
give the requisite potential 
difference. Vary the resist- 
ances B and B' till on clos- 
ing Ki and K2 in order, the 
galvanometer shows a mini- 
mum deflection. Until the 
balance is nearly completed 
it is better to insert in the 
shunt circuit containing the 
galvanometer and standard cells S a high resistance. If 
no current passes through the galvanometer the electro- 
motive force of the standard cells is equal to the poten- 
tial difference between the binding-posts of B'. Read 
now the voltmeter V. 

Then 

V=2I! ^+^' (for two standard cells), 




H 



I I I lllll I I 



Fig. 



where Vis the number of volts and B the electromotive 



206 ELECTRICAL MEASUREMENTS. 

force of the standard corrected for temperature. If the 
voltmeter is direct reading, the difference between V 
and the reading will be the error at that part of the 
scale. 

The voltage may then be changed and another bal- 
ance taken, continuing the process till the entire scale 
has been traversed. 



QUANTITY AND CAPACITY. 207 



CHAPTER V. 



QUANTITY AND CAPACITY. 

97. The Ballistic Galvanometer. — The quantity of 
electricity discharged through a galvanometer during a 
transient flow may be measured by means of the first 
swing of the needle, provided its period of vibration is 
sufficiently long to permit the passage of the discharge 
before the needle moves through an appreciable angle. 
Such a galvanometer is called a ballistic galvanometer. 

The general expression for a continuous current with 
any galvanometer is 

where QfS equals the magnetic field, Gr is the galvanome- 
ter constant, and 6 is the angular deflection. 

When the deflection is small, with any galvanometer 

G 
The present problem resolves itself into finding what 
function of the deflection must be multiplied by the 

constant -— - to give the quantity discharged through the 

Gr 

galvanometer. 

The maximum moment of the deflecting couple, due 
to a current J, is 

2mlia=8mia (Art. 6^2), 



208 



IJLECTBICAL ME A S UBEMENTS. 



where I is the half length of the needle and 81b its mag- 
netic moment, 2ml. The moment of a couple producing 

an ansfular acceleration — is K — , in which K is the 
^ dt dt 

moment of inertia of the movable system. 

Therefore 

dco 



8f]Ua = K 



dt 



The instantaneous value of I is 
current is constant. 



Therefore 



gma 



dQ 

dt 



K 



dQ 
dt ' 

dw 



for I. 



Q 



when the 



dt 



If ft) is zero at the instant when the circuit is closed, 
then integrating, 

3maQ = Kco (1) 

We must now obtain the expression for 
the energy of motion of the system at the 
instant when 6 becomes zero and place it 
equal to the work done in producing a 
deflection. The kinetic energy of a rotat- 
ing system in terms of moment of inertia 
and angular velocity is 

iKcoK 

Now, if the total work done on the 
needle is represented by the kinetic energy 
of the system as it passes through the 
position of zero deflection, that is, if 
there is no damping of any kind, then 
this energy may be equated to the work 
done on the needle against the force of control. If the 
impulse on the needle moves it from the position of 




5^ml 



Fig. 99. 



QUANTITY AND CAPACITY. 209 

equilibrium through an angle 0, the work done on it in 
moving its poles a distance Aa (Fig. 99) against the 
controlling force BSin on each pole is 2S8m. Aa. 

But Aa = I (1 — cos 6^. Hence the work done on 
both poles in producing a deflection 6 is 

2BSml (1 - cos (9) = BSdlb (1 - cos 6'). 

Therefore J^w^ ... g^^97S (1 - cos 6*). 

But from equation (1) 

Hence 

imaqy ^ ^^^ _ ^^^ ^ ^^^ ^_ e_ ^ 

K 2 

Solving, 

0= — \/ sm- = — \ .2sin-. (2) 

The time of a single vibration of the magnet is given 
by the equation 

&S3m 



'_/3JbG 
\ K 



'-^\ 



from which ^ ' 



^~^Jl 



&88m 

Substituting in (2), 

« = f .?...„?. ... (a) 

This is the full equation for quantity Avithout any 
damping coefficient. 

If 6 is small, sin 6 may be taken equal to - 6^ and 
2 2 

or the quantity is proportional to the first angular throw. 



210 



ELECTRICAL MEASUREMENTS. 



If T is the observed time of a single oscillation for 
an amplitude a, then the time for an infinitely small arc 
is given by the equation 

Table III. in the Appendix contains the corrections. 




Fig. 100. 



If d is the deflection in scale parts and a the distance 
between the mirror and scale, then 



^« = ^C-i?) 



For accuracy the value of To should be used in the 
above formulas for Q. 

A ballistic galvanometer complete is shown in Fig. 100. 



QUANTITY ANB CAPACITY. 211 

The astatic system, consisting of four bell magnets, is 
at the right of the cut. W is the soft iron ring nut 
which is employed as a magnetic shunt to adjust the sen- 
sibility of the whole astatic system. When it is turned 
up toward the poles ns the magnetic moment of this 
lowest magnet is diminished 

98. Correction for Damping. — A correction to the 
deflection 6 may be necessary on account of the damp- 
ing action on the needle due to setting the air in motion, 
and to the induced currents produced in the coil by the 
movement of the needle. 

If the deflection is small, so that we may write the 
angle for the sine of the angle, and if the damping also 
is small, we may write 

where 6 is the first deflection, and 6' is the following 
one in the same direction.^ Here the decrement of the 
first half-swing is taken equal to one-fourth the total 
decrement of the succeeding four half-swings ; or the 
decrement of the first quarter of a period is taken equal 
to one-fourth the decrement of the complete period fol- 
lowing. 

The logarithmic decrement of the motion is the Nape- 
rian logarithm of the ratio of any one amplitude to that 
which succeeds it after an interval of half a period. Let 
it be denoted by \. 

To apply it to the correction for damping, let Wj, n.,^ 

^L' Electricity, Mascart and Joubert, Tome 1, p. 558. 



212 ELECTRICAL MEASUREMENTS. 

%, etc., be successive scale readings. Then the ratio of 
one amplitude to the next following is 

P = 



n. — n., 
and loge/o = X. 

The constancy of the ratio /o, or of the logarithmic 
decrement X, means that the successive amplitudes 
decrease in a geometrical series. Let the differences 
between the successive scale readings, that is, the suc- 
cessive amplitudes, be denoted by ^i, «2, a^^ etc. 

Then a., = J:\ as= -^ = -1; a,^ = L . 

P PR' P"~' 

Whence loge«n = ^og^ai — (n — 1) log^/j, 
and loge«„i = loge^i — (wz — 1) loge/o, 

where the first equation applies to the n''^ amplitude and 
the second to the m'^. Subtracting the first equation 
from the second, 

loge^^ — loge^^ = (n — ??i) logeP := (^71 — m) X. 

Therefore 

x=^_ logy^-y 

n — m yJin/ 

If a^ is the first amplitude and a^ is the ?i'\ then 



n — l \a^/ 



If now a represent the first amplitude not diminished 
by damping, ai being the observed amplitude, then 

n — m for the two is _ , and 

A 

2. 

or 

-X + loge^l = lOgea. 



QUANTITY AND CAPACITY, 213 

But 

Now 



^' = ' + 1 + 1-2 + 



Therefore 



e*' =1 + Jx + 



and a = a^ ll +-Xj, omitting higher powers. 

If then X be determined by observing the first and ?^'* 
amplitudes and substituting in the equation 

the equation for quantity becomes 



fM^) 



^1, 



where 6^ is the first angular deflection,- and the damping 

is small. 

Example. 

Scale readings + 130, — 120, + 105, — 97, + 85. 

iloo- 13? _A 
4 °^° 85 ' 0.4343 



Hence A = i log^^ '^ . ^,-±^^ == 0.1062, 



and 1 + 1^=1.0531. 

^2 

The damping correction amounts to 5.3 per cent. 

99. Standard Condensers. — Standard condensers 
are made of tin foil interlarded with mica, and finally 
embedded in solid paraffin. The experimental deter- 

* AVilliamson's Differential Calculus, p. 62. 
^Maxwell's Electricity and Magnetism, Vol. II., p. 357. 



214 ELECTRICAL MEASUREMENTS. 

mination of the capacity of such condensers is more or 
less affected by conductivity and by absorption. The 
capacity with solid dielectrics is a function of the dura- 
tion of the charging. For a primary standard of capacity 
it is necessary to use a condenser with air as the dielec- 
tric, an instrument which Lord Kelvin calls an air- 
Leyden. The insulation resistance, which should be 
several thousand megohms, may be measured by one 
of the methods in Chapter III. ; and if any portions of 
a subdivided condenser are found to have faulty insula- 
tion, they cannot be used. The paraffin used by the 
best foreign makers has been known to contain traces 
of acid which attacks the metal embedded in it, and 
causes the insulation to deteriorate. When the top is 
clean and dry a good condenser should not lose an 
appreciable part of its charge in an hour. The influ- 
ence of absorption can be eliminated only by the appli- 
cation of the method of rapidly alternating charges and 
discharges. 

A subdivided condenser is usually made in the form 
shown in Fig. 101, in which one side of all the sections 
is connected to the brass bar marked Earth., and the 
other sides to the blocks J., ^, C, i>, E., as indicated by 
the dotted lines. When any section is to be used it is 
connected by a brass plug to the bar marked Condenser. 
The other sections may at the same time be completely 
discharged by connecting to Earth. For example, the 
condenser has a capacity of 0.3 microfarads when A^ B^ 
and C are connected to Condenser., J) and E being to 
Earth. It is evident that great care must be exercised 
in putting in the plugs, for the batter}^ applied may be 
short-circuited if plugs are inserted at both ends of any 
block. 



QUANTITY AND CAPACITY 



215 



The accuracy of a staudard condenser may be tested 
by comparing the different sections with one another 
when a second condenser is not available. Thus charge 
A by connecting to Condenser^ all the other blocks being 
joined to Earth, Then remove all plugs and divide A^s 
charge with B by connecting both blocks to Condenser, 
A and B should then have equal charges if their capaci- 
ties are equal. This can be determined by discharging 
first one and then the other through a ballistic gal- 





















EARTH 




"T"! 






f 1 




Ujj 




■ h 


~~^ 


•[•"i 




ri, 1 


9 


A 


i i 

'■T'' 


B 




C 




D 




E 


|6 






1 ' 




r--' 




■-■' 


L/-J 


r-' 


L^^ 






.05 




.05 .2 

CONDENSER 


.2 .5 

















Fig. 101 



vanometer and observing the throw. Use sufficient 
E.M.F. to get a satisfactory deflection. Next compare 
C and D in the same manner. Then charge J., -5, and 
C simultaneously, divide (7's charge with D, and aiscer- 
tain whether the charge of A and B together is equal 
to that of C and D separately. Finally, charge ^J., B, 
(7, and B together, and divide their charge with E. 
The discharge of E should then give the same throw 
of the galvanometer as that of the other four together. 
For this method the tops of the plugs should be well 
insulated. 



216 ELECTRICAL MEASUREMENTS. 

Any one of the sections may be made the basis of 
a comparison for the remainder. In every case the 
charges compared by the ballistic galvanometer Avill be 
very nearly equal. Hence, the deflections may be taken 
proportional to the charges without error ; and since the 
charges are proportional to the capacities, 



Hence 





' 0, d,' 


C,= 0, 


d^ 


3 between A and B. 


u 


^ + ^andia 


(,(, 


C and B. 


(i 


A-\-B-\- C+B'^indB, 



Then ^ = 0.05. 

B=OMa. 

C = 0.05(1 + a) 25. 

B = 0.05 (1 + d) 2hc. 

E= 0.05 (1 + a) (1 + 25 + 250 ^• 

100. Comparison of Capacities by the Method of 
Divided Charge. — The method of calibrating a stand- 
ard condenser described in the last article may be applied 
to the comparison of any capacity Oo with that of the 
standard C^. 

The standard is first charged by a potential difference 
which need not be known, but which must remain of 
fixed value. The charge Qi is then measured by dis- 
charging through a ballistic galvanometer. The stand- 
ard is again charged to the same potential difference, 
and therefore with the same quantity $i, and is then 



QITAXTITY AXD CAPACITY. 217 

connected for a fcAV seconds in parallel with the cable 
or condenser whose capacity is to be measured. The 
charge Qi di^ddes in proportion to the capacities of the 
two connected condensers. The charge remaining in 
the standard is then measured by the ballistic galvanom- 
eter. Call it Q. Then the charge in the condenser of 
unknown capacity Co is Qi — Q, and 



Whence 



Q 

For the highest accuracy 6\ should be equal to Cs • 
This may be demonstrated by the general principle of 
Art. 36. In this case we wish to find the partial 
derivative of Co with respect to Q. 



0, 


_Q.- 


Q 


c\ 


Q 




c,= 


r, ^^'- 


-Q 



The minus sign is used with the derivative, because 
C2 decreases as Q increases. The relative error is 

F^. ft 



For a minimum the denominator Q(^Qi — Q') must be 
a maximum, since Q is the variable and ft the constant. 
But § + ( ft — C*) = ft ? a constant ; and when the sum 
of two factors is a constant their product is a maximum 
when they are equal to each other, or when Q= Q^— Q. 
Hence for a relatively minimum error, 

This means that the charges must divide equally, or 



218 



ELECTRIC A L ME A S UREMENTS. 



6\ must equal 0.> . After a preliminary trial, therefore, 
adjust the subdivided standard condenser so that the 
capacity used shall be as nearly as possible equal to the 
capacity to be measured. 

The chief objection to this method lies in the fact 
that the two charges compared bear the ratio of about 
two to one. Hence, the observed deflections of the 
ballistic galvanometer must be corrected to render them 

proportional to sin - 6 instead of tan 26 (Table I.). 



c, 



c^ 



R, 



101. Comparison of Capacities by the Bridge 
Method. — Let the two condensers be placed in two of 
the arms of a Wheatstone's bridge, 
and two resistances in the other 
two (Fig. 102), the galvanometer 
joining the branches on either side 
connecting a capacity to a resist- 
ance. Adjust the resistances i^, 
and B, till the galvanometer shows 
no deflection on charging and dis- 
charging by means of the key K. 
When there is a balance the poten- 
tials of the points A and B remain 
equal to each other during charge 
and discharge. Hence the two con- 
densers, being charged AAdth the 
same difference of potentials, will 
contain quantities proportional to 
their capacities, or 

Q, o: 

But the quantities flowing into the condensers in the 



1 



Fig. 102. 



QUANTITY AND CAPACITY. 219 

same time are inversely proportional to the resistances 
Bi and i^o. Hence 

or a=C,§. 

1x2 

The resistances By and B. must be non-inductive 
and without capacity. It is desirable for accuracy that 
the two capacities should be nearly equal to each other, 
and that the resistances should be moderately large. 

The charge and discharge of long cables or of cables 
coiled in tanks is much retarded by absorption and elec- 
tromagnetic induction. Hence when the time constants 
of the two condensers compared are very different the 
bridge method may give a result largely in error, partic- 
ularly for rapid charge and discharge. To avoid this 
error the key K should be worked slowly. 

Example. 

Comparison of a subdivided condenser with one marked « 

microfarad, but found by an absolute determination to have a 
capacity of 0.3345 mf. 



Subdivisions. 


R, 


R., 


^2 


0.05 


1046 


7000 


0.0500 


0.05 


1042 


7000 


0.0498 


0.2 


4140 


7000 


0.1978 


0.2 


4151 


7000 


0.1983 



102. Comparison of Capacities by Gott's Method. 

— This is also a bridge method, but differs from the last 
one in exchanging the places of the galvanometer and 
battery. The arrangement is shown in Fig. 103. 

Two resistances B^ and B.^ are selected inversely pro- 



220 



ELECTRICAL ME A S UBEMENTS. 




portional to the supposed values of Ci and C,- The 
key Ki is then closed and clamped. After a few seconds 

key K2 is closed, and if any 
deflection of the galvanometer 
occurs, the condensers are 
discharged by opening Ky and 
closing K2 . After readjusting 
Ri or R.2 the operation is re- 
peated and continued till on 
closing K2 with the battery 
still in circuit no deflection 
is produced. 

Since the two condensers 
are connected in cascade they must contain the same 
quantity and C\ Vi = C-^T'l, where Vi is the fall of poten- 
tial over i?i, and K, that over i?2- 
Hence 

a Vi Ry 

The battery remains in circuit except during the dis- 
charge of the condensers. For highest accuracy the 
resistances should be quite large and the capacities 
equal. 

The galvanometer key should be well insulated, as 
well as the conductors leading to the condensers. It is 
not necessarv to insulate the battery. 



103. Correction for Absorption. — The last method 
furnishes a means of measuring the absorption of one of 
the condensers compared. Assume C\ as the one which 
absorbs a charge. Obtain a balance exactly as with the 
Gott method. The inverse ratio of the resistances will 
not be then the ratio of the true capacities. For, since 



QUANTITY AJSD CAPACITY. 221 

the same quantity Q has entered each condenser, while 
a portion q has been absorbed, the potential difference 
between the two sides of Oi is due to a charge Q — q^ 
while the potential difference of C^ is due to the charge 
Q. Then 

r,^^-^^, andF;=^, 

where F^ and V-> are the differences of potential between 
the terminals of Ri and R.j respectively when a balance 
has been obtained. From the two preceding equations 



Therefore, 



C.^V, q^ ^E, 5_ / -, , B, 

0^ Ti C,V,~E, C,E, 






where JE^ is the electromotive force of the battery. 

To find ^, with the key Ki closed adjust Ri and R^ so 
that the galvanometer shows a small deflection due to 
the discharge of a fraction of the charge of C^ on closing 
the key K2. This is effected by diminishing Ro slightly 
relative to Ri. 

Then open K.2 , break the circuit at K^ , and after a few 
seconds close K2 and observe the deflection. The gal- 
vanometer needle should now swing in the opposite 
direction to that observed before opening the battery 
circuit. If necessary readjust the resistances till the 
two opposite deflections are equal to each other. The 
quantity discharged through the galvanometer in either 
direction is then equal to q. 

To find now the value of ^, charge a condenser of 
known capacity with a known E.M.F. and discharge 
through the ballistic galvanometer. Let the deflection. 



222 



ELECTRICAL MEASUBEMENTS. 



corrected for clamping, be d-,^ and let the deflection due 
to g be c?i . Then 

Ct-2 

where C is the known capacity and E the known E.M.F. 



104. Comparison of Capacities by Thomson's 
Method of Mixtures. — This method takes its name 
from the process of mixing the charges of opposite sign 
of the two condensers compared in order to determine 

whether those charges are 
equal. C (Fig. 104) is a 
Pohl's commutator, which 
must be well insulated. 
When it is turned so as to 
connect the terminals of 
the batter}' with the inner 
coatings of the two con- 
densers, Ci and C.2^ they 
are charged with the po- 
tential differences existing 
between the terminals of 
the two resistances Ri and 
R.2 respectively. When 
the commutator is turned 
the other way, the two charges of opposite sign mix. 
To ascertain whether the}^ are equal and completely 
neutralize each other, the key K is then closed and any 
residue remaining in either condenser is discharged 
through the galvanometer G. The resistances B^ and 
R.2 should be large and the capacities about equal. The 
electromotive force should be as large as the resistance 



V J^ 




1 1 l| 

L- R, ^-T—' R. --^ 



A 
Fig. 104. 



QUANTITY AND CAPACITY. 223 

boxes will safely permit, especially for the final adjust- 
ment, since onl}' the residue of the two charges remains 
to affect the galvanometer. 

The point A is sometimes grounded. This is essen- 
tial when the capacity of a cable is to be measured. 
The core of the cable is then connected to the com- 
mutator and the earth is the outer coat. High insula- 
tion of the rest of the apparatus is essential. 

Example. 

To compare a special mica condenser C2 with a Marshall con- 
denser Ci of 0.3345 microfarad capacity. 

590 340 0.3345 0.1928 

1400 807 0.3345 0.1928 

105. Discharge of a Condenser through a High 
Resistance. — When a non-absorbing condenser leaks 
through a high resistance i^, the fall of potential is 
expressed by the equation 

V= V,e ^^ (Art. 51), 

in which V\ is the initial potential or charging electro- 
motive force, and V is the potential after the condenser 
has been leaking t seconds through a resistance R. If 
potentials are plotted as ordinates and the times of leak- 
ing as abscissas the curve will be exponential in form. 

Since the quantity held by a condenser of capacity C 
is proportional to its potential, we may also write 



We also have 



Q=Qoe 



R^i ^ ' 



^ log,-| ^ log, |x 2.303 



224 



ELECTBICAL MEASUREMENTS. 



as the resistance through which the condenser leaks, 
expressed in terms of common logarithms and the deflec- 
tions of the ballistic galvanometer employed to measure 
the charges. 

The actual curves obtained by experiment will differ 
from the theoretical exponential ones because of the 
complication introduced by absorption. So also the re- 
sistance computed from observations made at different 
time-intervals of leakage will not be constant, but will 
increase with the time. 

The apparatus may be set up as in Fig. 105, in which 
^ is a charge and discharge key. When the lever h is 
brought in contact with a the condenser is charged by 
the battery B. If the lever h is thrown over to c the 

whole charge is at 
. ^ once passed through 

the galvanometer G-. 
This gives the de- 
flection do. Then 
charge again and 
place the lever mid- 
way between a and 
c for five minutes or 
more, the time de- 
pending upon the 
insulation resistance 
of the condenser. If 
that is too high to 
permit of frequent observations, a resistance of about 25 
or 30 megohms, if available, ma}^ connect the two sides 
of the condenser. At the end of the observed time of 
leaking, the lever b is again made to touch c, and the 
deflection corresponding to the charge remaining in the 




QUANTITY AND CAPACITY. 225 

condenser is observed. Charge again and proceed in 
the same way, increasing each time the period of leak- 
ing till a sufficient number of observations have been 
secured. 

It is obvious that all parts of the circuits, including 
the galvanometer and the battery, must be highly insu- 
lated. The deflections, or the corresponding quantities, 
may then be plotted as ordinates and the periods of 
leaking as abscissas. 

106. Residual Discharges. — For the purpose of 
studying the residual charge it is advisable to experi- 
ment with a cable of sufficient capacity and with an 
insulation which constitutes a dielectric of large absorb- 
ing power when the cable is immersed in water. A 
cable of high insulation resistance should be selected. 

Proceed as follows : Charge the cable with an electro- 
motive force of 50 to 100 volts for several hours. It will 
often continue to absorb a charge for twenty-four hours. 
Discharge it through a low resistance by closing the key 
for a very short interval. This is best accomplished by 
using the pendulum apparatus (Art. 58) and setting 
three keys so that the ffi'st one opens the charging cir- 
cuit, the second discharges through the low resistance, 
and the third insulates the cable. Let it stand insu- 
lated for five seconds and then discharge through a 
ballistic galvanometer. Next charge again to the full 
by applying the same electromotive force as at first for a 
period about twice as long as the cable has been left 
insulated. This is done by resetting the keys on the 
automatic pendulum device in the proper order. 

Then again discharge througli the low resistance, and 
increase the time of standino- insulated to ten seconds. 



226 ELECTRICAL MEASUREMENTS. 

passing the residual cliarge as before through tlie gal- 
vanometer. Recharge for about double the time the 
residual charge occupied in coming out, and repeat 
the observations with increasing intervals of insulation. 
Finally, plot the deflections (or quantities) and the cor- 
responding periods of insulation. 

Example. 

A coil of insulated wire, which had been in a tank of water 
for 15 days, was charged b}^ a storage battery of 73 volts elec- 
tromotive force for several hours. The length under water was 
997 feet, its capacity 0.075 microfarads, and its insulation resist- 
ance 400,000 megohms. 

The keys on the charge and discharge apparatus were so set 
that the cable was discharged through a low external resistance 
for about \ second. The insulation periods ranged from one 
second to two minutes. The following are the data of the ex- 
periment : 

Intervals Mean Quantities in Rise in volts 

in seconds. Deflections. microcoulombs. during the intervals. 

1 19.4 0.357 4.76 

2 22.8 0.420 5.60 

3 25.4 0.467 6.23 
5 28.3 0.521 6.95 

10 34.9 0.632 8.43 

15 37.9 0.697 9.29 

20 40.2 0.740 9.87 

25 43.1 0.793 10.57 

30 . 45.5 0.837 11.16 

40 48.0 0.883 1177 

50 50.5 0.929 12.39 

60 51.9 0.955 12.73 

90 57.1 1.051 14.01 

120 62.6 1.152 15.36 



QUANTITY AND CAPACITY. 



227 



It Tvas necessary to shunt the galvanometer with the ^^ shunt, 
because without it after the fifteen-second period the deflection 
became too large. Its constant was then 0.0184 microcoulomb per 
mm. deflection. The observations are plotted in Fig. 106. 



40 



20 

























^ 














^^ 




-^ 








O 




f^ 




^' 
















ij 


y^' 






















/ 
























r 


































Sec 


:>nds 













10 20 30 40 50 60 70 80 90 100 110 120 

Fig. 106. 



107. To measure the Absolute Capacity of a Con- 
denser — First Method.' — When a quantity of elec- 
tricity Q is discharged through a ballistic galvanometer, 



Q 



a 






9 is the first angular throw. 

Let A represent the constant ^ , and for put — , in 

which d is the deflection and a the distance of the scale 
from the mirror, both in millimetres. Then 

lira 



Stewart and Gee's Practical Physics, Part 11. , p. 407. 



228 ELECTRICAL MEASUREMENTS. 

If a condenser of capacity C be charged with an 
E.M.F., E, then 

Q = EC. (2) 

From (1) and (2) 

C=A^S^±MA, ... (3) 

If now we use the same battery to produce a steady 
deflection cli through a resistance R, including that of 
the battery and the galvanometer, then 

l=^£ (^) 



for small deflections. 
Therefore, 



A _ 2a 

E~Rdr 



Substitute in (3) and 

r_ T(l + ^X) d ... (5) 
itR dx 

In practice fii^st determine d by charging the condenser 
with an electromotive force ^, as in Fig. 88, discharging 
through the ballistic galvanometer, and notice the deflec- 
tion or first swing d. 

Next, find the time of a single vibration, correcting 
for reduction to an infinitely small arc. 

Third, determine R and dy. R must be a high 
resistance, and probably the y^^ shunt will need to be 
used with the galvanometer. Increase R until the 
deflection is within the proper limits. Then if R^ is 

the external resistance, h that of the batterv, and ^^ 

9 + ^ 



QUANTITY AND CAPACITY. 



229 



that of the galvanometer and shunt in parallel, the total 
resistance in circuit will be 



f^ + 



But snice the shunt is used, the equivalent resistance for 
the current measured is 

■ ff + S 



Ii = 



fn^ + h 



+ 



9 



Finally, substitute in equation (5). If B is in ohms 
C will be in farads. 

108. Absolute Capacity of a Condenser — Second 
Method. — This method rests upon the production of a 
steady deflection of the galvanometer by a succession of 
rapid discharges through it from the condenser. If the 
rate of discharge is a large number of times the frequency 
of oscillation of the galvanometer needle, the effect of 
these discharges in pro- 
ducing a deflection is 
the same as that due to 
a current numerically 
equal to the quantity 
of the discharges a 
second. 

The apparatus may 
be set up as in Fig. 
107. JT is an automatic 
device for charging 
the condenser and dis- 
charging it through the galvanometer at an iinvarying 
rate. The tuning-fork with the attachment described in 
Art. 92 may be employed. 




Fig. 107. 



230 ELECTRICAL 31EASUBEMENTS. 

If n be the immber of discharges per second, C the 
capacity of the condenser, and ^ the charging electro- 
motive force, then for one discharge q = JEC^ and for n 
discharges 7iq = jiEC. This quantity is equal to the 
current I^ which will produce the same deflection. 

If di is the deflection in scale parts, corrected by Table 
II. for [)roportionality to tan ^, then 

md, = nEC (1) 

where m is a constant equal to the current corresponding 
to a deflection of one scale part. 

Next connect in series the same battery, the gal- 
vanometer, and a high resistance i?i, the galvanometer 
being shunted with a resistance s. Then if do is the 
deflection, corrected as before, 

md-, = <; )■ — — • ... (J) 

Divide (1) by (2) and 

^ = uRC '-±^. 
do s 

R is the total resistance of the circuit, neglecting the 
internal resistance of the battery. 

Therefore, 6^=='^^ ^ ' 



I2 nR s -\- g 

109. Absolute Capacity of a Condenser — Third 
Method. — The condenser whose capacity is to be meas- 
ured is placed in one of the branches of a Wheatstone's 
bridge (Fig. 108). One side of the condenser is alter- 
nately connected to aS' for charging and to R for dis- 



QUANTITY AND CAPACITY. 



231 



secoijcl by means of a vibrating 



charging n times a 
plate P, or a tuning- 
fork (Art. 92). The 
condenser is thus 
charged and dis- 
charged n times a 
second. During the 
charging of the con- 
denser a part of 
the charge passes 
through the galva- 
nometer in the oppo- 
site direction to the 
steady current flow- 
ing when the con- 
denser is f nlly 
charged and while it is discharging. 

The resistances are varied until a balance is obtained 
as in the use of the Wheatstone's bridge for the meas- 
urement of resistance. Then if the resistances of the 
several branches are represented by the small letters in 
the figure, 

\ { a + c ^ g) {a + b + d) - a^la 

nC = ? L .1 

j (a -f & + t?) (a + c) - a (a + c?) j j (« -h (?) (« 4- c + ^) - a ( a + c) j 

In practice it has been found unnecessary to use the 
complete formula. Where a and h are small in com- 
parison with c?, ^, and d^ we may write 





M. J. Thomson, in Phil. Trans., 1883, Part III., p. 707; E. T. Glazebrook, 
in PJdl. Mag., 1884, Vol. 18, p. OS.. 



232 ELECTRICAL MEASUREMENTS. 

This approximate formula may be demonstrated as 
follows : • 

The quantity required to charge the condenser equals 
the product of its capacity and the maximum value of 
the potential difference between D and B which is 
reached when the condenser is fully charged. Assuming 
that the time required to charge the condenser is a very 
small fraction of the period of the fork, we may sup- 
pose a steady current flowing through the galvanometer 

for - of a second, followed by a momentary rush through 

n 
it in the opposite direction of that part of the charge 
which goes through the branch g. The galvanometer 
needle will appear to stand still in its zero position if 
the total quantity passing through the galvanometer is 
algebraically zero. The period of the galvanometer 

.1 

must be large in comparison with - . 

The value of the steady current through d is 



h + 



a (^ + (/) 



a + c^ g 

if E is the E.M.F. of the battery. Put R for the 
resistance 

a-^c+g 
Then the steady current through g is 
E a 

R' a+ c-^g' 
These currents cause a fall of potential between D 
and ^ of 

R\:~^arVc+g)' 



QUANTITY AND CAPACITY. 233 

Hence the total quantity required to charge the con- 
denser to this potential difference n times a second is 



n 






Neglecting self-induction, the portion of this charge 

passing through the galvanometer is ~ times the 

a + c + g 

whole, and this discharge is balanced by the steady cur- 
rent through the galvanometer in the opposite direction 
for the rest of the period. 
Therefore, 



R\ a+ c+ g/\a + c + gj 



E 



R a-\-c + g' 
Hence, 
nC= — 



if a is negligible in comparison with c. 

Example. 

Measurement of the absolute capacity of a Marshall one-third 
microfarad condenser : 



n 


9 


a 


c 


d 


C 




32 


13,720 


5 


1000 


467 


0.331 




32 


13,720 


1 


1000 


95 


0.329 




32 


13,720 


3 


1000 


281 


0.330 




32 


13,720 


2 


1000 


187 


0.330 


ohm 














Mean, 


0.330 


mf 

B.A. unit. 



234 ELECTRICAL MEASUREMENTS. 

The resistances were in B.A. units. The dimensional 
formula of a capacity is L~^T^^ while that of a resistance 
is IjT~^. Hence, the unit of time remaining the same, 
any change in the unit of resistance is directly as a 
length, while the change in the unit of capacity is 
inversely as a length. Therefore, the resulting change 
in the numeric of a capacity, measured in terms of a 
resistance, will be directly as a length, or directly as the 
unit of resistance. The international ohm is 1.01358 
B.A. units. Hence, 0.330 microfarad measured in B.A. 
units equals 

0.330 X 1.01358 = 0.3345 mf. 

The charge and discharge was effected by means of a 
large Koenig fork, and its rate was measured by means 
of a device based on electrolytic action. Its rate both 
immediately before and immediately after the balance 
was found to be just 32. 



SELF-IXDUCTIOy A^^D MUTUAL INDUCTION. 235 



CHAPTER VI. 



SELF-INDUCTION AND MUTUAL INDUCTION. 

110. Preliminary Relations. — The electromotive 
force of self-induction in any circuit or part of a circuit is 
the product of its inductance L and the rate of change of 
the ciUTent. If the resistance is strictly non-inductive, 
then L is zero and there is no self-induced electromotive 
force. If the circuit or coil contains no magnetic mate- 
rial and has no iron within or about it, then X is a 
constant, and the electromotive force of self-induction is 




Fig. 109. 

proportional to the rate of change of the current. The 
phase of this electromotive force is then a quarter of a 
period behind that of the current, when the latter is 
simple harmonic. 

Let an alternating current, following the simple har- 
monic law, be represented by the heavy sine curve I of 
Fig. 109. Then the induced electromotive force due to 
its variations may be represented by the thin line II. 
This is also a sine curve, since the differential coeffi- 



236 



ELECTRICAL ME A S UBEMENT8. 




Fig. 110. 



cieiit of a sine function is itself a sine function. When 
the current has reached its maximum value at A, the 
electromotive force has its zero value, because at that 
instant the change-rate of the current is zero ; but when 
the current passes through its zero value at B, its 
change-rate is a maximum and the induced electromo- 
tive force has its great- 
est value. The electro- 
motive force therefore 
reaches its maximum 
■-^^ value one-quarter of a 
period later than the 
current, and the two are 
said to be in quadrature. 
The effective electro- 
motive force producing the current in accordance with 
Ohm's law must correspond in phase with the current 
itself. We may therefore represent the maximum effec- 
tive and induced electromotive forces by the two adjacent 
sides of a right triangle^ (Fig. 110), where ah is the 
effective electromotive force, and he the inductive elec- 
tromotive force; the hypotenuse ac is therefore the 
maximum impressed electromotive force applied to the 
circuit. Since the current is in the same phase as ah, it 
must lag behind the impressed electromotive force by an 
angle (/>. This angle becomes zero when L is zero. 
Self-induction therefore explains the lag of the current 
behind the impressed electromotive force. 

The instantaneous value of an alternating current fol- 
lowing the simple sine law is 

i = I sin 6 = 1 sin ^irnt. 



^ Carhart's University Physics, Part I., p. 36, 



SELF-INDUCTION AND MUTUAL INDUCTION. 237 

where I is the maximum value of the current, and n is 
the number of full periods per second. Hence 27rn is 
the angular velocity co. 

Therefore, i = I sin cot. 

Then, X --== ZwJcos cot. 

dt 

The maximum value of this induced electromotive force 
is LcdL Therefore in the triangle of electromotive 
forces, if the base ah is the maximum effective electro- 
motive force, producing a current I through a resistance 
i^, by Ohm's law it is equal to BI. Also hc^ the max- 
imum inductive electromotive force, is LcoL Conse- 
quently the hypotenuse ac equals I*^ W + Hcd^ or 

J5'=7ViF+XV. 

Therefore, ^ 



VW+L'co- 



The expression (i?' + L'co^)^ is called the impedance. 
Also, tan (i — ^ . 

In these equations /and E may represent either the 
maximum values of the current and electromotive 
force, or the '' square root of the mean square " values. 
The latter are those measured by all the practical current 
and pressure instruments which are operated by forces 
varjdng as the square of the current and electric pressure 
respectively. Such are the electrodynamometer and the 
electrostatic voltmeter. 

111. To solve for the Current when the Circuit 
contains both Self-induction and Capacity. — If the 
electromotive force applied follows the simple law of 



238 ELECTRICAL MEASUREMENTS. 

sines, its value for any instant is e = ^ sin cot. This 
applied electromotive force equals tlie vector sum of the 
effective electromotive force producing a current, the 
electromotive force of self-induction, and that due to 
the charge of a condenser in series with the resistance. 

Then, JE sin cot = Ri + L '-' + J^ . 

at C 

The last term is the electromotive force introduced by 
capacity. From the definition of capacity the potential 

c 

But Q=/idt. Hence r= i^^. 

It is entirely valid to assume a general solution of the 
above equation and then fuid the constants. Since the 
applied electromotive force is a sine fmiction of the time, 
it may be assumed that the current also will be a sine 
function if the circuit contains no iron. The general 
equation for the current may then be written 
{= Jc sin (cot — <j)~). 

The angle ^ is introduced to express the lag of the 

current behind the applied electromotive force. Then 

di 
Zi ~ — LhoD cos (wt — ci)). 
dt ^ ^^ 



fidt h . , ,^ 1 

I = — TV cos (cot — (ii)} 

C Geo 



1 Strictly speaking, this equation should be written 

■7?^ = -/cOS(a,^-^)+^, 

in which ^ is a constant of integration. It will however be easily seen that 
the value of A is zero, as the raaximum and minimum values of 

must be numerically equal, which is true only when A is zero. 



SELF-INDUCTION AND MUTUAL INDUCTION. 239 



Substituting in the equation of electromotive forces, 

k 

E sin cot = Rk sin (cot — (^) + QLkco — -) cos {cot — <^). 

Ceo 

Since this equation is generally true, it is true when 
the angle (wt — (^) equals zero and when it equals - . 

A 

k 

In the first case E sin <^ = Lkco .... (a) 

Ceo 

In the second case E cos (^ — Rk (5) 

Squaring (a) and (5) and adding, 

E'=R'k' + k' 



■(- 1) 



and k = 



Therefore, / = ^ sin (cot — cf)}. 

To find the angle of lag, divide (a) by (6) and 

J- 1 

l/CO — -— - 
Ceo 

tan (/) = ^ • 

While self-induction causes the current to lag behind 
the impressed electromotive force, capacity tends to give 
to it a lead ahead of the electromotive force. The one 
will neutralize the other when 

Ceo 

The fraction -— - is the impedance due to capacity alone. 
Ceo 

It may be expressed numerically in ohms. 



240 



ELECTRICAL MEASUREMENTS. 



If the circuit contains self-induction but not capacity, 
then the third term in the equation of electromotive 
forces drops out and 



where / and E are either maximum values or the square 
roots of the mean squares, as measured by an electro- 
dynamometer. 

If the circuit contains capacity but no self-induction, 
then 

7: " 



V 



R. 



Q-ai' 



Further, if the resistance of the circuit is zero, 

This last equation furnishes an independent method 
of measuring the capacity of a condenser. 

112. Measurement of the Capacity of an Electro- 
static Voltmeter.^ — The voltmeter is 
ih'st employed to measure the potential 
difference e between the alternating 
mains. A non-inductive graphite resist- 
ance of several megohms is then joined 
in series with the voltmeter. It will 
now indicate a smaller potential differ- 
ence 60- This potential difference is 
one-quarter of a period behind the 
charging current, while the potential 
difference e^ between the terminals of 
the graphite resistance agrees in phase 

^Dr, Sahulka, in the Proceedings of the Chicago International Electrical 
Congress^ p. 379. 




SELF-INDUCTION AND MUTUAL INDUCTION. 241 

with the current, since this resistance r is non-inductive 
and without capacity. 

Hence (Fig. Ill) e = s/~~f^2. 

Therefore e^ maybe computed and /equals — . Then 

since I also equals 27rne.20^ 



e., 



1 
27rnr 



Example. 

The alternating current had 2500 full periods per minute. 

Hence oj=2nn = 262. 

The table gives the results with a Kelvin multicellalar volt- 
meter. The values of r are in megohms, the potential difference 
in volts, the current in millionths of an ampere, and the capacity 
in millionths of a microfarad. 



r 


e 


ei 


69 


i 


C 


11.05 


207.2 


69.2 


195.3 


6.26 


122 


20.78 


207.6 


108.3 


177.1 


5.21 


112 


33.16 


207.6 


138.6 


154.6 


4.18 


103 


41.90 


207.9 


153.4 


140.3 


3.66 


99.6 


52.40 


208.0 


166.7 


124.4 


3.18 


97.6 



The capacity was greater for the higher values of 62 than for 
the lower ones, because the movable system is deflected so as to 
increase the capacity of the instrument as an air condenser for the 
higher readings. 

113. Measurement of Capacity by Alternating Cur- 
rents. — Employing small letters for the square root of 
mean square values, 

6-2(0 

where eo is the potential difference between the two sides 
of the condenser. If i is expressed in amperes and c. in 



242 



ELECTRICAL ME A S UREMENTS. 



volts, C will be in farads. Let the condenser be put in 
series with a graphite resistance, about numerically 

equal to the impedance of the condenser -— expressed 

Ceo 

in ohms. By means of an electro- 
static voltmeter measure the potential 
difference between the terminals of 
the graphite resistance and between 
those of the condenser. Call the 
former e^ and the latter e., • Then e^ 
agrees in phase with the current, 
while 6-2 differs from it in phase some- 
what less than 90° if the condenser 
has a solid dielectric. Measure also 
Fig. 112. g, the potential difference between the 

mains. Then since i equals -, 

r 

e., 27rnr 
The angle of lag a may be calculated from Fig. 112. 

e~ =z ei + el+ ^eieo cos a. 

Whence 




cos a 



e'> 



2e^e, 



The energy in watts absorbed by the condenser is 

IV = eoi cos a. 

In an air condenser, where a is 90°, the energy absorbed 
by the condenser during the charging is equal to that 
restored to the circuit in the discharge, or the positive 
work done equals the negative. In condensers with 
solid dielectrics energy is absorbed in excess of that 
given out and the condenser heats. 



SELF-INDUCTION AND MUTUAL INDUCTION. 243 

E^zample. 

To measure the capacity of a nominal Jj microfarad made by 
Elliott Bros, the smaller electrostatic voltmeter of Art. 95 was 
employed. The alternator had 10 poles and made 1,632 revolu- 
tions per minute. 



r 


e 


6] 


eo 


G 


i 


« 


OJ 


16700 


105.7 


83.5 


62.25 


0.093 


0.005 


88" 13' 


0.0097 



The capacity of a condenser with a solid dielectric is smaller 
when measured with alternatino- currents than with direct ones. 



114. Impedance Method of measuring the Coeffi- 
cient of Self-induction.' — The value of the coeffi- 
cient of self-induction of a coil of known resistance M 
.may be found by passing through it an alternating- 
current and measuring the potential difference between 
its terminals by means of an electrostatic voltmeter. At 
the same time the current through the coil must be 
measured by an appropriate ammeter. Then 



1= 



U 



V R + L' 



the 
the 



where E is the measured potential difference, I 
current, R the olimic resistance of the coil, and L 
inductance in henrys. 

The term Lw is now called the reactance. The resist- 
ance must be measured independently, and w is obtained 
from the speed of the dynamo and the number of poles. 
Thus a small bipolar machine, making 3000 revolutions 
a minute, gives for n a value of 50, and for co or 27r>^ 



^Nichol's Laboratory Manual of Physics, Vol. II., p. 109. 



244 



ELECTRICAL MEASUBEMENTS. 




314.2. The value of L may then be found by substitut- 
ing the values of E^ 7, i^, and &> in the equation for the 
current. 

Draw a right triangle (Fig. 
113) with the three sides 
equal to resistance, reactance, 
LcD and impedance respectively, 
and measure the angle of lag 
(/). Compute the time con- 
stant of the coil := . If the re- 

sistance of the coil is large, the result may be vitiated 
by its static capacity.^ 

The value of L found by this method depends upon 
an ammeter and a voltmeter reading. It may be made 
to depend upon voltmeter readings alone. 

115. Three-Voltmeter Method of measuring Induc- 
tance.^ — A non-inductive resistance Ri (Fig. 114) is 
placed in series with the coil of resistance R2 whose 
inductance L2 is to be measured. An alternating cur- 



R R L 

^ _ ^ 

Fig. 114. 

rent is then sent tln-ough the circuit, and it may be 
measured by the ammeter J. as a check. Three volt- 
meter readings as nearly simultaneous as possible are 
taken — E the total potential difference between the 

1 Electrical World, July 13, 1895. 

-Nichol's Laboratory Manual, Vol. II., p. 113. 



SELF-INDUCTION AND MUTUAL INDUCTION. 245 

terminals of the whole resistance, JEi between those of 
Hi , and Uo between those of M2 . 

Then draw a triangle OBA (Fig. 115) with the three 
sides equal to the three voltmeter readings, or the read- 
ings reduced to 
volts. Produce 

OB to (7, mak- ^^ 

ing CA a right ^^ / 

triangle. Then v^^^^ / 

AC \% equal to ^^ Az • U&^i 

L2(oL It may be ^^ / 

taken directly ^^ / 

from the figure, q^^— —^ J- ?f J— 

and L2 may then Pig ,|5_ 

be found from 

the known values of the frequency n and the cur- 
rent. BO is the electromotive force producing the 
current /through Bo^ and OA the electromotive force of 

seK-induction. It is evident that /equals ~ , since B^ 

\ 

is non-inductive. Besides the three electromotive forces, 

we must therefore measure either J or B^. 

If the coil surrounds an iron core, the inductance 
should be measured for different values of the current. 
It will be found to decrease as the core becomes satu- 
rated. The currents may then be plotted as abscissas 
and the inductances as ordinates. 

116. Comparison of the Capacity of a Condenser 
with the Self-Inductance of a Coil.^ — The four resist- 
ances in the arms of the Wheatstone's bridge (Fig. 116) 
are Q^ P, i?, ^S^. When the battery circuit is closed, the 



1 Maxwell's Electricity and Magnetism, Vol. II., p. 



246 



ELECTRICAL MEASUREMENTS. 



potential difference at the terminals of E causes a cur- 
rent through it and at the same time charges the con- 
denser O. The potential difference rises as the condenser 
receives its charge, and therefore the current through M 
requires a definite time-interval to rise to its final value. 
The current through the coil Q will increase from 

zero to its max- 
-^ imum value in a 

precisely similar 
way on account 
of the counter 
E.M.F. of self- 
induction. Both 
the condenser 
and the coil have 
a time constant, 
and the effect of the condenser in delaying the current in 
one branch may be made to offset that of the coil in the 
other, so that the rise of potential at F may be the same 
as at H. In that case no current will pass through the 
galvanometer. We have to determine the conditions 
under which the potential at F remains equal at every 
instant to that at H. 

Let X and z be the quantities which have passed through 
P and M respectively at the end of the interval t after 
closing the circuit. Then x — z will be the charge of 
the condenser at the same instant. 

The potential difference between the two sides of the 




condenser is by Ohm's law H 



— 1, since ^~ is the value of 
at at 



the current. Therefore 



x — z 



EC 



ydZ 
Jt 



(1) 



SELF-iy DUCT ION AND MUTUAL INDUCTION. 247 

Let 1/ be the quantity traversing Q in the same time 
t. Then the potential difference between A and IT is 
equal to that between A and F when there is a balance 
and no current flows through the galvanometer ; or 

^ dt cie dt ^ ^ 

The first member consists of the effective E.M.F. pro- 
ducing a current and E.M.F. of self-induction. The 
sum of the two is the potential difference between A 
and H. 

Since there is no current through the galvanometer 
the quantity passing along HZ must be the same as that 
along AH^ or y' — y. Therefore 

^dy j^dz .3. 

dt dt ^ ^ 

since the potential difference between F and Z is the 
same as that between H and Z, when no current flows 
through the galvanometer. 

From (1) dx_dz^^^^dH^ 

dt dt dt 

the rate at which the condenser is charged. 
Substitute in (2) and 

dt dt- y dt dtj 

From (3) -^ — —. . Substituting in the last equa- 

^ ^ dt S dt ^ ^ 

tion and Q -^-{- L - ^^ = P { i?(7 + — | . 

S dt Sdf V df dt 



( 

Multiply by S and integrate and 



QBz + LR '^4 = PRSC^ + PSz, 
dt dt 



248 ELECTRICAL MEASUREMENTS. 

This is the equation of condition that no current shall 
pass through the galvanometer. 

The condition for a steady current with a Wheats 
stone's bridge is 

QR = PS. (5) 

Hence the condition that no current shall traverse the 
galvanometer when the battery circuit is opened and 

closed is 

1 = ^^- (^) 

— and RO Sixe called the "time constants" of the coil 

Q 

and the condenser respectively. If by varying P and R 
the bridge can be adjusted so that no current traverses 
the galvanometer on opening and closing the battery 
circuit, as well as when it is kept closed, then the two 
" time constants " are equal and 

L = QRO. 

To show that a time constant ^ is a time, since a resist- 

R 

ance has the dimensions of a velocity, and a capacity is 

the square of a time divided by a length, we have from 

the equation -^=R0 (calling the coefficient of self-induc- 

tion L' to distinguish it from a length X) 
J-, . L_L T_ ™ 

Also i'=('4Y- -~ = i. 



■=©■?= 



SELF-INI)UCTIOy AND MUTUAL INDUCTION. 249 



or self-induction is a length. The unit of induction is 
the henry and equals 10' cms. It varies directly as the 
ohm. 

If C is in microfarads the value of L from the equa- 
tion above will be 
multiplied by lO"*^ to reduce to henrys. 



a million times too large and must be 



117. Anderson's Modification of Maxwell's Method.^ 
— In the preceding method of Maxwell a double adjust- 
ment must be made in order to effect a balance. First, 
one of the branches P has to be adjusted for a balance 




Fig. 117. 



with a steady current. Then, in order to obtain a bal- 
ance when the galvanometer circuit is closed first, the 
resistance R will have to be adjusted. This necessitates 
a fresh adjustment of P, and so on. Anderson's modifi- 
cation of Maxwell's method is designed to facilitate the 
adjustments. 

Suppose a balance has been obtained for steady cur- 
rents by closing ^^ before jSTq (Fig. 117). This balance 
will not be disturbed by introducing the resistance r 
between F and iV. Adjust r therefore till the galvanom- 

1 FMl. Mag., Vol. XXXI., 1891, p. 329. 



250 ELECTRICAL MEASUREMENTS. 

eter shows no deflection when K2 is closed before K^, 
The potentials at H and N then remain equal to each 
other. Let x be the quantity which has fiow^ed into the 
condenser at the time ^, and z the quantity which has 
passed through FZ. Then x -\- z has passed through AF. 
Then if C is the capacity of the condenser, since the fall 
of potential from ^ to -^ is the same by the two paths, 

we have 

7-> (XiZ X . (XiX /--I •\ 

^dt=o^'Tt ^^> 

Also since N and H m^ust be of the same potential, 

-c-'% (2) 

Further, the change of potential from A through F 
to iV^ is the same as from A to H. Hence 

^dx -p/dx dz\^Qdy -^d^ 
dt \dt dt) ^ dt dt' ^^ 

Substituting from (1) and (2), 

C " A- p\d^ \-^ ( ^ 1 d^\ — ^ X L dx 
^ ^dtR\0dtl~~S'0'^'Sddt' 

This equation expresses both conditions necessary for 
a balance with variable currents. For steady currents 

R S' 



Hence the other condition is found by equating the 
dx 
di 



coefficients of -^ ; or 



B so 



SELF-INDUCTION AND MUTUAL INDUCTION. 251 

This condition o-ives the formala 

If r is zero, L = OPS = CQR, which is Maxwell's 
formula. 

To apply the above equation for X, first obtain a 
balance in the ordinary way, and then adjust r and, if 
possible, C till there is no deflection of the galvanometer 
needle on working Ki with Ki closed. 

For sensitiveness of the final adjustment it is desirable 
to make B and aS' large, and r small. Since Q is usually 
small, P will also be small. 

Example. 

Calibration of the Standard of Inductance, 

1. For a balance with steady currents, 

P= 13.27 ohms. i? = 125.2 ohms. 

g = 10.6 '' fif=100 

When corrected for temperatm-e, Q-\- S =^111.1 ohms; PX S 
= 1337. 

2. For a balance with variable currents, 

C=. 0.335 mf. 



r 


Nominal value of 


Calculated value of 


in ohms. 


inductance. 


inductance. 


124 


0.005 


0.0051 


253 


0.010 


0.0099 


390 


0.015 


0.0150 


525 


0.020 


0.0200 


660 


0.025 


0,0250 


798 


0.030 


0.0301 


925 


0.035 


0.0349 



118. Russell's Modification of Maxwell's Method.^ 
— Connect the coil exactty as in Maxwell's method and 
balance for steady currents. Then if the galvanometer 

^ London Electrician, Maj 4, 1894. 



252 ELECTRICAL MEASUREMENTS. 

key be closed first, there will be a throw of the needle 
when the battery key is closed ; and if the battery key 
be opened first the throw of the needle will be the other 
way. Now connect the condenser, which should be a 
subdivided one, as a shunt to the branch R. The effect 
will be to reduce the throws of the needle. Use different 
values of the condenser capacity, one giving a throw in 
one direction on opening or closing the battery circuit, 
and the other a throw in the other direction. Then by 
interpolation find the capacity which would reduce the 
deflection to zero. This capacity, substituted in the 
equation L = QR (7, gives the desired inductance L. 

Example. 

To yneasure the, Self-Inductance of Two Coils. 
The bridge consisted of special non-inductive resistances. 
i? = 131.7 ohms. >S = 131.2 ohms. 

^ = 25.88 ohms -f- resistance of the coil. 
The coils consisted of 450 turns in three layers each, the 
smaller having a mean diameter of 3.3 cms., the larger, 4.0 cms. 
The larger coil could be slipped over the smaller one. 

1. The smaller coil. 

Q = 30.05; R= 131.7. 
With C= 0.45 mf., the deflection was -}- 15 scale parts. 

C = 0.5 " " " " —25 '^ 

To balance, G = 0.47 mf . 

Therefore, 

L = 0.00000047 X 30.05 X 131.7 = 0.00186 henry. 

2. The larger coil. 

^ = 31.15; R = 131.7. 
With C =0.6 mf., the deflection was + 65 scale parts. 

" (7 = 0.7 " " " " — 15 

To balance, C = 0.68mf. 

Therefore, 

L = 0.00000068 X 31.15 X 131.7 = 0.00279 henry. 
The condenser was a microfarad subdivided into .5, .2, .2, .05, 
.05 mf. 



I 



SELF-IXDUCTION AND MUTUAL INDUCTION. 253 



119. Rimington's Modification of Maxwell's 
Method/ — In this method one side of the condenser 
is connected to F (Fig. 118), and the other side to a 
point jV, which 
can be shifted 
along so as to 
vary r withont 
any change in 
the resistance R 
of that branch. 
In this arrange- 
ment the dis- 
charges through 

the galvanometer, due to the discharge of the condenser 
and the self-induction of the coil, are in opposite direc- 
tions and equal, when both balances have been secured. 
Let y be the current flowing in the arms Q and S^ when 
it has reached its steady value, and x that in P and R. 

Let both keys be closed and then let Ki be opened. 
The quantity of electricity which passes through the 
galvanometer, due to self-induction in ^, is 




Ly 



R+ S _ Lya 



Q + 



a(R + S} a + R + S F+Q+G-a 

a + R^-s 



This is the integral of the current between the limits 
and y. The quantity passing through the galva- 
nometer from the discharge of the condenser is 



Cxr 



F+Q 



Cxr'-h 



p _u ^' -u ^(-P + ^) 0- + P+Q R + s+ah 



Phil. Mag., Vol. XXIV., 1887, p. 54. 



254 BLECTBICAL MEASUliEMENTS. 

This discharge passes while the current through r falls 
from X to zero. 

These quantities pass through the galvanometer in 
opposite directions, and if there is no deflection, 

Lya _ Cxr-b 



But 

and 



F+ Q+ aa B + S+ ab 

Lya ^ Ly (^R+ S ')^ 

F -^ Q+ G-a e 

Oxrb Oxr' (P + Q^ 

B + s+ab~ c 



Hence Ly (R + S} = Cxr (P + 0. 

Now5 = 4. Therefore i .:^= |. ^,= |, 
y P y R+ S P R-\-S R' 

since PS— QR. Hence 

L= Cr'-^. 
R 

If r = P, we have Maxwell's formula, 
L= OQR. 

The resistance must be such that r can be adjusted 
without changing the value of P after a balance has 
been obtained for steady currents. The double com- 
mutator, illustrated in Fig. 47, may be used in this 
method when suitable adjustments of the two commu- 
tators are made. 

The condition L= -— ^ may be obtained directly 

from Maxwell's equation P= ORQ. When no deflec- 



SELF-iyBVCTION AND 2IU.TUAL INDUCTION. 255 

tioii of the galyanometer is observed on opening the 
battery circuit, a certain quantity of electricity, coming 
from the condenser, must pass through the brancli aS'. 
If one of the terminals of the condenser is moved along 
H to the point iV, the fraction of the charge passing 

through /S'will be decreased in the ratio of—-; and as 

the total charge will be decreased in the same ratio 
because of the lower potential to which the condenser is 
charged, the quantity passing through S on the dis- 

charge will be reduced in the ratio of — -. Consequently, 

if the same quantity is to pass through S as in Max- 
well's method, the capacity of the condenser must be 

7?- 

increased in the ratio of — , . Whence it follows that 

r 

R 



120. Comparison of Two Coeflacients of Self- 
induction.* — The double commutator of Fig. 47 may 
]3e used for this pur- 
pose to increase the ^^'^ 
sensibility. The four 
points of the bridge 
(Fig. 119) are con- 
nected with the 
double commutator 
exactly as in Fig. 49. 

Let Ri and R2 be 
inductive resistances with coefficients L^ and L^, and 
let Rz and R^ be inductionless resistances. Then if 




Maxwell's Elec. and Mag., Vol. II., p. 



256 ELECTRICAL MEASUREMENTS. 

H. and H^ be adjusted to give a balance with a steady 
current, a balance will also be obtained with varying 
currents when 

Li Mg 

The rate of rotation of the commutator must not be 
too great to permit the currents to reach their steady 
values between consecutive reversals. 

The equation may be demonstrated as follows: Let 
ii be the current through AC and 2*3 that through AD 
(Fig. 119), at the same instant t after closing the circuit, 
or after reversal. Then, since no current traverses the 
galvanometer when a balance has been obtained, Zj and 
H are also the currents through CB and DB respec- 
tively. The difference of potential between A and C is 
the same as between A and I) ; also the fall of potential 
from (7 to ^ is the same as from D to B. Hence 

Biii + A-t!'==^3«3, 
at 



Whence, 



at 



at at 

But BiB^ = BoBo^ is the condition of a balance with a 
steady current. The other condition for a balance with 
varying currents is therefore 

ju^B^ = IjoBg.) 

~ = — 
L2 B^ 

If one of these coefficients, as i, , is a standard of self- 



SELF-INDVCTlOJ^r AND MUTUAL INDUCTION. '257 

induction, the equation gives the value of the other. 
Such a standard is shown in Fig. 120. It contains two 
coils without iron joined in series, one of them fixed and 




Fig. 120. 



the other movable about a vertical axis. The self- 
induction of the tAvo depends upon their relative position, 
and the scale at the top is graduated to read in milli- 
henrys. Since the self-induction of the standard is 
variable, a balance can often be obtained for variable 
currents by changing the relative position of its two 
coils. Its resistance, however, is only about ten ohms ; 
and if the ratio of its smallest inductance to that of the 
coil to be measured is greater than that of their relative 
resistances, a balance can be effected onl}^ by adding 
non-inductive resistance in series with the standard. 



258 



ELECTBICAL MEASUREMENTS. 



Incandescent lamps in parallel or in multiple series are 
convenient for this purpose, since it is not necessary to 
know their resistance. 

Alternating currents and an electrodynamometer may 
be employed with advantage in this method (Art. 60). 
The entire current should pass through the field coil, 
and the suspended coil should take the place of the 
galvanometer, as in Fig. 50. 

Example. 

To compare the Two Coils of Art. 118 with the Standard of In- 
ductance. 
The standard was inserted in the arm i?i, together with an 
additional non-inductive resistance, the latter being added in 

order to increase the ratio -^, so as to brino' the induction in 
the arm B^ within the limits of the standard. 



Coil. 


^2 


R\ 


Zj (Standard). 


L-2 


Smaller. 

Larger. 

Two opposed 
in series. 


4.160 
5.245 
9.4 


S 36.15 
/ 49.37 
( 36.15 
\ 49.37 
S 141.6 
\ 167.5 


0.0161 

0.0219 

0.0195 

0.02665 

0.0152 

0.01795 


0.001853 

0.001845 

0.002829 

0.002832 

0.00101 

0.00101 



121. Niven's Method of comparing' Two Self-In- 
ductances.^ — The inductance of Ri is to be compared 
with that of R^ (Fig. 121). First connect Ri in a Wheat- 
stone's bridge with three non-inductive resistances Ro , R5 , 
and Rq and obtain a balance for steady currents. Then add 
the inductive resistance R^ in series with Ro and balance 
again for steady currents by adding a proportional non- 
inductive resistance to Ri . Finally connect E and F by 

^ Phil. Mag., Sept., 1877. 




SELF-INBUCTION AND MUTUAL INDUCTION. 259 

means of the resistance i^r, and vary it till the galva- 
nometer shows no 

deflection on mak- c 

ing and breaking 
the battery cir- 
cuit. 

Call the quan- 
tity of electricity 
which has passed 
throng h each 
branch of the cir- 
cuit q^ with the 

proper subscript, at the time t after closing the circuit, 
and let Q with the corresponding subscripts represent the 
quantities for the several branches when the current has 
reached a steady state, after an interval T, reckoned from 
the time when the circuit is closed. Then the current is 

represented by — ^ and this is zero for each branch when 

t is zero. It is also zero for the two cross branches i^7, 
i^s, when the steady state has been reached. 

The potential difference between C and D at any time 
t is the same by the four paths. Hence 



Fig. 121. 



at dt 



dqc, _ -pfdq 
dt 
j^dq, ^ j^d^ _ J, dq., _ 



d-q, 



dq^ 



d-q^ 



E ^ -\- L — ^ = i?4 -^ + X 

dt dt'- dt df 



dt 



dt' 



dt 



at at- at 






dt- dt 

. . . . (1) 



Integrating between limits ^ = and t—T^ we have all 
the terms zero when t is zero; and putting 7i, Z,, J^, 



260 ELECTRICAL MEASUREMENTS. 

etc., for the maximum values of the several currents, 
corresponding to t= T^ the equation above becomes 

R,Q, -R,Q, = BsQs + LJ,= R,Q, + LJ, + E:Qr + L,I, - 
R,Q, = B,Q, + LJ, + R,Q, - R,Q,- LJ, - R,Q, (2) 

But since the galvanometer shows no deflection, both Qs 
and is are zero, and 

B,Q, = B,Q,ov^ = ^; . . . . (S) 

and since a balance exists for i^i, ito, with i^-,, R^^^ also 
for Ri + Rg and R2 + R^ with R^^ i^^, it follows that 

Ri Ro R- ^ 

Ro R4 Re, 

also since ft; = ft and ft, = ft;, there being no flow 
through the galvanometer, we have 

R,Q,-RsQs= 0, 
because by substituting in (3) 

^3_ ft 

Ri ft 

I^ is also zero. It follows from equation (2) that 

LJ, + R:Q:=0 (4) 

and R,Q, + LJ,-R,Q,-LJ, = 0. . . (5) 

Further ft; = ft, - fta and ft, -|- ft, = ft + ft , 
or ft>=ft+ft4-ft. 

Substituting in (4) and (5) 

LJ, + R:Qr-R:Qs=0. ... (6) 

R2Q3 + R2Qi - R2Q1 + LJ, -R,Q,- LJ, - . (7) 
Multiplying (6) by (R, + R^ and (7) by R^ and adding. 



SELF-INDUCTION AND MUTUAL INDUCTION. 261 



(i^i + Bo + B:} LJ, - B:LJ, - B:B, Q, + B,Bj Q, ^ 0. (8) 

Therefore from (8) 

Li B^ + Bi + B: li Bi + Bo + B: B^ 

L\ B- i\ B-j Bq 

7? 7? 7? 

The ratio — ' may be replaced by -— ^ or by — ^ . 
Bti B2 B^ 

Esiample. 

Coinx)arison of the Inductances of the Two Coils of the Last 
Fxamjyle. 

In the branch i?i was put the larger of the two coils with an 
additional non-inductive resistance, so that i?i was 31.1 ohms. 

Arm i?2 was another non-inductive resistance of 25.9 ohms. 
In i?4 was the smaller coil (4:. 16 ohms), balanced in Bs by a part 
of the non-inductive resistance of Fig. 80. R5 and Bq were 
formed by a slide wire bridge, the point B being the sliding 
contact. 

The first balance was obtained by moving the contact at B, 
and the second by adjusting the resistance Bs. i?? was a resist- 
ance box and was 200 ohms for a balance with variable currents. 
Then 

ii ^ 57 4- 200 81.1 ^ ^ .^g 
Li 200 * 25.9 — -^ • 

From the last experiment, 

122. Mutual Induction/ — Mutual induction is the 
induction taking place between adjacent circuits. The 
coil or circuit in which the inducing current is made to 

iNichol's Laboratory Ma7iual, Vol. I., p. 242. 



262 ELECTRICAL MEASUBEMENTS. 

vary is called the primary, and the circuit acted on 
inductively is the secondary circuit. 

Let P be the primary and S the secondary (Fig. 122). 
The primary coil is connected in series with a battery B^ 
a variable resistance i^, and an ammeter A. In the 




circuit of the secondary are connected a resistance r and 
a ballistic galvanometer Gi. 

Firsts observe the throw of the galvanometer needle 
when ^is opened and closed. At the same time measure 
the steady current flo^ving through the primary. Then 
reduce R and repeat the observations, keeping the resist- 
ance of the secondary circuit constant. The resistances 
R and r should be so adjusted that the series of deflec- 
tions of the ballistic galvanometer may vary from the 
smallest that can be accurately read to the largest that 

the scale will allow. The readings may be corrected for 

a 
proportionality to sin - . 

A 

Finally plot the primary currents as abscissas and the 
corrected deflections as ordinates. The resulting curve 
should be a straight line passing through the origin, or 

§oc7, (a) 

where Q is the quantity of electricity discharged through 
the secondary, and I the current in the primary. 

Second^ to determine the relation between the quantity 
of electricity which flows in the secondary circuit and 



SELF-INBUCTION AND MUTUAL INDUCTION. 263 

the resistance of that circuit, observe the throw of the 
galvanometer when the primary circuit is closed and 
opened for several different resistances in the secondary. 
Then plot the deflections as ordinates and the reciprocals 
of the resistances as abscissas. The result will be a 
straight line through the origin. Hence 

<2ai, (5) 

in which R is the resistance of the secondary circuit. 
Combining (a) and (5), we have 

The constant Mi^ defined as the coefficient of mutual 
induction, or the mutual inductance, of the two coils. 
It is the electromotive force induced in the one coil 
while the current varies in the other at the rate of one 
ampere per second. 

The value of M depends on the geometrical form 
and winding of the two coils and on their relative 
position. 

Tliird^ Q may be measured in coulombs by finding the 
constant of the ballistic galvanometer, using a condenser 
of known capacity and a standard cell. If, further, / is 
measured in amperes and R in ohms, then M in the 
above equation will be expressed in henrys. 

Example. 

I. The quantity is proportional to the primary current. 
The table contains the results of an experiment. In the third 
column the deflections are corrected so as to be proportional to 

2 sin i^^ 



264 



ELECTRICAL MEASUREMENTS. 



Current in Primary. 


Observed Deflections. 


Corrected Deflections. 


1.470 


31.2 


30.78 


1.295 


27.5 


27.22 


1.159 


24.5 


24.31 


0.982 


20.9 


20.77 


0.722 


15.2 


15.15 


0.627 


13.2 


13.16 


0.518 


10.7 


10.68 


0.420 


8.8 


8.78 


0.330 


6.9 


6.89 


0.214 


4.5 


4.49 


0.113 


2.3 


2.3 



II. The quantity is inversely proportional to the resistance of 
the secondary circuit. 



Resistance of 


1 


Observed 


Corrected 


Secondary. 


B 


Deflections. 


Deflections. 


7000 


.0001429 


35.6 


35.15 


17000 


.0000588 


14.3 


14.26 


27000 


370 


9.1 


9.08 


37000 


270 


6.6 


6.59 


47000 


213 


5.2 


5.19 


57000 


175 


4.25 


4.24 


87000 


115 


2.78 


2.77 


107000 


0934 


2.2 


2.2 



The first and third columns of the first table and the second 
and fourth of the second table are plotted as coordinates in 

































^ 


























_^ 


^ 


y^ 


^ 






















^^ 




<f 




^ 




















y 




y6 
























J^ 


^ 




"n 






















^ 


^ 


























^ 


^ 


■^ 
























^ 


^ 


^ 





























t 16 18 
Fie. 123. 



23 24 26 2S 



SELF-INDUCTION AND MUTUAL INDUCTION. 265 



Fig. 123. 
orio;in. 



The result in both cases is a straight line throuo;h the 



123. Comparison of Two Mutual Inductances. ^ — 
Let ^1, Ao (Fig. 124) be the two coils whose mutual 
inductance M^ is 

Q 



AAAAAA-^fAAAAA 




Fig. 124. 



to be compared 

with the mutual 

inductance il/j^ of 

the coils A:i, A^. 

The coils J.3, A^ 

are placed in the 

required relation 

to each other, 

while Ai and Ao must be at such a distance from A^ and 

A^ that there is no mutual inductance between Ai and 

Ai , nor between A.2 and As . The coils are joined in 

series as shown. Then the resistances of the branches 

containing Ai and A^ must be varied by the addition 

of non-inductive resistances till the galvanometer shows 

no deflection on closing and opening the key K, The 

sensibility will be increased by the use of the double 

commutator. When a balance has been obtained, 



if,. 






The theory is as follows : Write the potential differ- 
ence by the three paths between P and Q and place 
them equal to one another. Then 



dii 



(H-i 



di\ 



Mu^ -Will -R,h=M,,^^^ 
dt dt dt 



L,'^-B,h^Bz + L^. 



dt ' ' ■ dt 

R^ 2, and L are the resistance, current, and inductance 



MaxwelVs Flee, and Mag., Vol. II., p. 364. 



266 ELECTRICAL MEASUREMENTS, 

for the circuit through the galvanometer from P to Q. 

If we integrate this equation from i^ = to ^4 = J, or 
to the time of the establishment of a steady current, then 

Mul— Ufdh — R-ifiidt = J/10/— L^/dh — Rifkdt = Rfidt + Lfdi. 

But L?fdiz and Lyfdix are both zero, since the currents 
z'l and U are zero when i^ is zero, and vv^hen it is J, a 
maximum. Also, since the adjustment of resistances is 
so made that there is no integrated current through the 
galvanometer, the last two terms are both zero. Hence 

M^- R^i^ = MJ- R./i^dt = 0, 

and 31, J = Rjhdt ; MyJ= R.fi^t, 

Therefore 

M^_Ri fhdt 
M:^~Rs ' Jhdt' 

But since there is no integrated current through the 
galvanometer 

fi.dt^fhdt. 

Hence —^ = -— ^ 

M^ R,' 

After a balance has been obtained the resistances Ri, 
Rg may be measured by means of a Wheatstone's bridge. 
It is assumed in this discussion that Li and X3 are both 
constants. 

124. Modification of Maxwell's Method of coni- 
paring Mutual Inductances. — Let the resistance, self^ 
induction, and current through coil ^1 (Fig. 125), 
including the galvanometer, be represented by i?i, Xi, 
ii , respectively ; and let the same quantities for the coil 
J-sbe denoted by R-, Xo, and 4. The resistances are to 



SELF-IXDUCTIOW AND MUTUAL INDUCTION z6 i 



be varied till the galvanometer shows no deflection on 
working the key 
K. Let the cur- 
rents through ^^2 
and Ai be 4 and 
ii and their final 
steady values I2 
and li. Let H be 
the resistance of 
the branch AB^ 
and S that of 

PQ. When the currents in A2 and A^^ have reached 
their steady value 




Fig 125. 



Ii=l2 



R + R2 



Express the potential difference between P and Q by the 
three paths for any instant, and place their values equal 
to one another. Then 



' dt dt 



B^^ = M, 



''It 



dt 



Sii— jSi^ 



The electromotive force by the JLg branch is arranged to 
be opposed to that of the Ai branch. 

Integrate from ^ = to t = T when the steady state 
has been attained in the battery circuit, and 



M,Jo - LJdir - RJixdt = M,J, - LJdis 
S/hdt - Sf{,dt. 



RJhdt = 



But Lifdii, Rifiidt^ L-^fdi^^ Sfiidt are all zero when a 
balance has been obtained, or when the galvanometer 
shows no integrated current through it when the circuit 
is opened or closed, or on reversing if the double commu- 
tator is used. Since the current is zero when t — and 



268 ELECTRICAL MEASUREMENTS. 

when t — T, the sum of the increments dii equals that 
of the decrements — dii . Also the galvanometer shows 
the integrated current I'l to be zero. So also the integral 
ydis is zero because 4 is zero both when t = and when 
the steady state of i^ has been attained. Hence we may 
write 

31, Jo = M,J, - Bsfyh = - sfXdt. 

^ 

Therefore, Ikdi — — 1J:L. 
Remembering that J =l2 ^ — - , 

XL 

Hence, ^^^^^ + ^2 S __S E+E, 



31^ E jS-E, E jS-E, 

If S is infinite, that is, if the branch P § is open, 
31,, _ E + E. 
M,, E ' 

E^ E.^ Er,. and aS' must all be measured after the adjust- 
ment has been made for no deflection of the galYanometer. 

125. Carey Foster's Method of measuring Mutual 
Inductance/ — The principle of the method is as fol- 
lows : Let a constant battery be connected in series 
with one of the coils P, a known resistance i?, and a key 
K. Let a ballistic galvanometer and another resistance 
E^ be connected in series Avith the other coil S. Then if 
I be the steady current through P, 31 the mutual 
inductance, and r„ the resistance of the circuit through 
S. E' and the galvanometer, the quantity of electricity 

J Phil. Mag., Vol. XXIII., p. 121. 



SELF-INDUCTION AND MUTUAL INDUCTION. 269 



passing through the galvanometer on closing or opening 

MI 

the circuit will he Q = ^ — (Art. 122). 

Next suppose the galvanometer removed from this 
circuit and put in series with a condenser of capacity C\ 
connected as a shunt to the resistance B. On closing or 
opening the battery circuit the quantity of electricity Q^ 
passing through the galvanometer will be Q^ = IB 0. 
By combining these two equations it is possible to find 
the relative values of and M. It is better however to 
connect the apparatus as shown in Fig. 126, so that the 
charge and dis- 
charge of the con- 
denser, and the 
currents generated 
at the same time 
in S by mutual 
induction are in 
the same direction 
through (7, B\ and 
iS, If the resist- 
ances B and B^ 
and the capacity C are adjusted until there is no 
deflection of the galvanometer, the time integral of the 
galvanometer current until the steady current is 
reached will be zero, and the time integral Qr of the 
current from C through B' and S^ multiplied by the 
resistance r of the same path from U around to A, will 
be exactly equal to the time integral MI of the 
electromotive force of mutual induction in the coil S. 
The time integral of the electromotive force of self- 
induction will be zero. 




Fig. 126. 



270 



ELECTEICAL MEASUREMENTS. 



Therefore, Qr = ML But Q = IE 0. 

Hence M= CRr. 

The author of the method says that in order that the 
galvanometer current may be zero at every instant 
during the establishment of the steady current, it is 
essential that the coefficient of self-induction of the coil 
S should be equal to the coefficient of mutual induction. 
Under this condition it is possible to replace the galva- 
nometer by a telephone. 

Example. 

Small Induction Coil. — No iron core. Resistance of second- 
aiy, 194 ohms. Capacity of condenser, 4.926 microfarads. The 
secondary coil could slide endways remaining coaxial with the 
primary. The following are the results with the centres of 
the tvvo coils as nearly coincident as possible : 



R. 


r. 


Rr 

(C.G-.S. -units). 


15 


194 + 217 


6165 X 1018 


14 


+ 247 


6174 


13 


+ 282 


6188 


12 


+ 322 


6192 


11 


+ 367 


6171 


10 


+ 423 


6170 


9 


--490 


6156 


8 


--576 


6160 


7 


--688 


6174 


6 


+ 835 


6174 



Meau value of ^^— 6172.4 X IQi*. 
C 

Hence 
ilf = 4.926 X 10-'^ X 6172 X 10^^ = 3.0403 X 10', or 0.0304 henrys. 

In the same way the values of Mwere obtained for the same 
pair of coils with the secondary displaced endways through 
various distances. The following results are given in Professor 
Foster's paper : 



SELF-INDUCTION AND MUTUAL INDUCTION. 271 



Distance between 
centres of coils. 


Value of JL 


Distance between 
centres of coils. 


Value of M. 


0.55 


304.0 X 105 


8.55 


97.3 X 105 


1.55 


294.2 


9.55 


71.1 


2.55 


270.5 


10.55 


49.7 


3.55 


246.4 


11.55 


33.0 


4.55 


215.9 


12.55 


23.3 


5.55 


187.8 


13.55 


16.5 


6.55 


158.4 


14.55 


12.35 


7.55 


127.2 


15.55 


9.48 



These values are represented graphically in the curve of Fig. 
127, where the ordinates denote values of M and the abscissas 
distances between the centres of the coils. 



30 
28 
.26 
24 
22 
20 
18 
16 


>5k 


































\ 


































\ 


































\ 
































\ 


V 
































\ 


































\ 
































s 


\ 




















t 












\ 




















12 


^ 














\ 
































\ 


V 
















8 


















\ 


































\ 














4 
2 























\ 


































V 


>^ 




















Dist 


ctnce 


inC 


ms. 








^3- 




-O 



7 8 9 
Fig. 127. 



10 11 12 13 14 



16 



272 



ELECTRIC A L ME A S UBEMENTS. 



It is of interest to note that this curve is of the same form as 
that of Fig. 57. The mutual induction affecting the coil S 
depends upon the number of lines of force passing through it at 
different distances from the primary coil P. In the same way the 
force deflecting the needle of the tangent galvanometer depends 
upon the magnetic field due to the coil at the several positions of 
the needle. The tangents of the deflections therefore follow the 
same law of variation as that of the mutual inductance at different 
distances. 

126. To compare the Mutual Inductance of T"wo 
Coils with the Self-Inductance of One of Them.^ — 
Let the coil of resistance By and self-inductance L be 
mcluded in one branch ^6' of a Wheatstone's bridge 
(Fig. 128) whose other branches are non-inductive. 
The other coil of the pair is put in the battery branch, 

and is so connected 
that the current 
flows in opposite 
directions through 
the two coils. The 
self-inductance of 
the coil P therefore 
produces an electro- 
motive force oppo- 
site in direction to 
that due to the mu- 
tual induction M be- 
tween P and Q, and the one may be made to balance 
the other. 

The resistances Ri, Bi, B^, and B^ are to be adjusted 
till there is a balance for steady currents. Then we may 
get rid of transient currents through the galvanometer 




Fig. 128. 



Maxwell's Flee, and Mag., Vol. II., p. 365. 



SELF-INDUCTION AND MUTUAL INDUCTION. 273 

by altering Ro and H^ in such a way that their ratio 
remains constant. There will then be neither transient 
nor permanent currents through the galvanometer. 

Let the current from A to C be h , and that from A 
to 2), ?o . Then the current through Q will be ii + ^2 . 
The potential difference between A and C will be 

dt \dt dtj ^ ^ 

The potential difference between A and D is i^s^. 
Since a balance is maintained between 6^ and D 

■i^.,4 = i^,-. + i§-i^(f + §). . . (2) 

But if R>., i?3, and ^4 are inductionless resistances, 
I{42 = R\ii (3) 



Hence 



dt \dt dt) ^ ^ 



From (3) di^ _ Ri dii 

'dt~^,' 'dt 

Therefore from (4) L = m(i+ ^\, . . (5) 



C ^ t)' 



The double adjustment of R^ and i^4 may be avoided by 
joining A and B by J?-. Beginning with an adjustment 
in which the electromotive force due to self-induction is 
slightly in excess of that due to mutual induction, the 
latter may be augmented by diminishing the resistance 
R-; till a balance is obtained for transient currents. This 
addition does not disturb the balance for steady currents. 

Then the current through Q will be r'l + u + i- , and 



' dt \dt dt dt) 



0. . . C^) 



274 ELECTRICAL MEASUREMENTS. 



But h= ^^ and ^^ = ?1±A^ . ^V 
zV Bi + Bs dt B^ dt 



L^ = M(^+ ^1^1+ ^1+^3 ^\ 

^7 



From (6) 

^^ ** \dt ' Bo dt ' Bj ' dt 

This last method may be further improved b}^ trans- 
ferring the battery and key to the branch Bj. Then 



or 



L = m{^' + ^A (8) 



To demonstrate this relation it will be seen that equa- 
tion (5) is equivalent to 

L = M^ (9) 

This equation is true for all arrangements. In the 
last arrangement we need only find the ratio y . It is 

^" ^ — '^ . Substitute in (9) and equation (8) is the 
Be 

result. 



3IAGNETISM. 275 



CHAPTER yil. 



MAGNETISM. 



127. General Properties. — Iron is not the only 
magnetic substance, for nickel, cobalt, and liquid oxygen 
are also very conspicuously magnetic ; and probably 
there is no substance which is not susceptible to some 
extent to magnetic influence. In permanent magnets it 
has been noticed that there is a certain line through the 
centre of inertia which always takes a definite direction 
when the magnet is freely suspended at this point. This 
line is called the magnetic axis. In most localities this 
axis takes an approximately north and south direction, 
in the northern hemisphere the north-seeking end and in 
the southern hemisphere the south-seeking end pointing 
downward. In a simple elementary magnet the ends of 
the magnetic axis are called poles. In larger magnets 
the poles are not so definitely located. They might be 
defined as the centres of magnetic action resulting from 
the actual magnetization. In general they lie on the 
magnetic axis near its ends. 

Until within a few decades the magnetization was 
considered as residing on the surface of the magnet near 
the ends, while the middle portion of the magnet was 
considered to be without influence. Since the time of 
Faraday the conception of lines of magnetic force and 
induction has to a considerable extent supplanted that of 
the poles. These lines of induction are closed curves. 



276 ELECTRICAL MEASUREMENTS. 

The positive direction along them is by convention from 
the south-seeking or negative pole to the north-seeking 
or positive pole within the magnet, and vice versa with- 
out. Whenever these lines of induction meet a magnet, 
they tend to enter it by the negative and leave it by the 
positive pole. Magnetic action, from the point of view 
of lines of induction, goes on just as though these lines 
were stretched elastic cords mutually repelling one 
another. In polar language the same state of affairs is 
expressed by the law : Like poles repel and unlike poles 
attract one another with forces proportional directly to 
the product of the strength of the poles and inversely to 
the square of the distance separating them. 

For certain purposes the conception of polar action at 
a distance is more convenient; and as the above law 
does not contradict actual experiment, we may avail 
ourselves of it, whenever it may be convenient to do so, 
without invalidating the results. 

128. Strength of Pole and Strength of Field. — By 
convention we define as unity, a pole Avhich repels an 
equal pole at a distance of one centimetre with the force 
of one dyne. 

Strength of field at a point may be defined as the force 
exerted on a unit pole placed at that point. It is also 
the flux of magnetic force per square centimetre at that 
point. If this flux of force is represented by lines of 
force, the number of lines per square centimetre should 
equal the numerical value of the flux and of the strength 
of field. In a uniform field the lines of force are parallel 
straight lines. If a magnetic pole of strength m be con- 
sidered as located at a point 0, the strength of field at 
all points on the surface of a sphere of unit radius with 



MAGNETISM. 277 

as its centre will be numerically equal to the pole 
strength, and there will be m lines of magnetic force per 
square centimetre of surface. There will be therefore 
in all ^irm lines from a pole of strength m. The letter 
dS is generally used to designate strength of field. 

129. Intensity of Magnetization. — When we are 
dealing with a magnet whose magnetization is solenoidal,^ 
all lines of force pass from one end to the other without 
entering or leaving at the sides. In such cases the poles 
are at the ends and the intensity of magnetization S 
equals the strength of pole m divided by the area of the 

pole S, or J=- . 

130. Magnetic Moment. — If a solenoidal magnet is 
placed at right angles to a uniform magnetic field of 
strength BS^ the moment of the couple tending to turn it 
into parallelism with the field is BSml ; if BS is unity 
the moment of this couple is called the magnetic 
moment of the magnet, and is designated by 8T6 > or 
ml = 3JE. As the volume V of the magnet equals IS, 

it follows that S= ; or intensity of magnetization 

equals magnetic moment per unit of volume. If the 
magnetization of the magnet is not solenoidal, S will 
not be uniform throughout the magnet and the mag- 
netic moment will not be equal to ml, unless by I we 
mean a distance shorter than the length of the magnet 
so chosen that 3Tb shall equal ml. The magnetic mo- 

1 Solenoidal is derived from the Greek word meaning pipe-shaped. The 
idea conveyed by the word is that the flow of magnetic induction is confined 
within the magnet, just as the flow of water is confined within a water-pipe. 
In all cases the flow is parallel to the sides of the solenoid. 



278 ELECTRICAL MEASUREMENTS, 

ment, however, in aii}^ case will be equal to the moment 
of the turning couple, when the magnet is placed across 
a unit field so that its action on the magnet is greatest. 

131. Strength of Field within a Long Solenoid. — 
Suppose an indefinitely long prism (or c}4inder) to be 
uniformly and closely wound with many contiguous 
turns of insulated wire carrying a current of electricity, 
the direction of the wire being at every point as nearly 
as possible perpendicular to the edges of the prism. By 
reason of symmetry the resultant field within the coil 
at points indefinitely distant from the ends must be 
parallel to the edges (or to the axis in the case of a 
cylinder). Approaching the ends, the resultant field 
will begin to weaken and there will be a scattering of 
the lines of force. At points indefinitely distant from 
the ends all lines of force are therefore parallel, the 
equipotential surfaces are equidistant planes perpendic- 
ular to them, and the field is uniform. Consequently, to 
fuid the value of the field for all points it is only neces- 
sary to fuid it for any one. Suppose there are n turns 
of wire per centimetre of length, each carrying a current 
I; suppose a unit north pole carried one centimetre 
along the lines of force ; each of the ^tt lines of force 
proceeding from this pole will cut n turns of wire, thus 
producing an electromotive force in the solenoid of 47r??., 
and the work done on the pole will be ^ttuI ; conse- 
quently the force opposing the movement of the pole 
will be equal to the strength of the field, and &S — ^TrnL 
J is here in C.G.S. units each equal to 10 amperes; con- 
sequently if I is expressed in amperes, E'-fS = . 

If, instead of a prism, the form on which the wire is 



MAGNETISM. 279 

woiiiicl is described by the revolution of a closed plane 
curve about an outside axis in its plane, and if the wire 
wound about it for each turn coincides as nearly as pos- 
sible with the generatrix, the resultant field will at every 
point lie along circumferences described about this axis. 
The intensity of the field at any point within will be 
4:7rnl as before, 7i being computed along the correspond- 
ing circumference. The field 88 will, as a consequence, 
not be uniform, but it will be absolutely solenoidal, there 
being no ends to cause a scattering of the flux. 

132. Magnetic Induction. — Let us suppose a long 
iron bar placed in a uniform magnetic field of strength 
BS so as to be parallel to the lines of the field. The 
portion of the bar which is distant from the ends will 
have its lines of induction parallel to its axis. Suppose 
the portion of the iron included between two adjacent 
planes normal to the axis to be removed. Let the flux 
of magnetic force passing across this crevasse per square 
centimetre be 6B? and let the pole strength per square 
centimetre of the faces be c9. Then 6B =4:7rJ-\- BS-, for 
each unit of pole strength will furnish a flux of ^tt in 
addition to the pre existent flux of &S per square centi- 
metre. Even when the above conditions are not ful- 
filled, the relation d^> =^ ^tt S -\- &S is true in a vector 
sense. Li the cases with which we shall deal, cfS will 
be parallel to cB either in the same or in the opposite 
direction ; then Qq = 47rc^ + EfS- 

This flux of force in the crevasse continues as a flux 
of induction inside the iron. In the crevasse it may 
be called indifferently a flux of force or of induction. 
Consequently lines of induction are continuous through- 
out the magnetic circuit. 



280 ELECTRICAL MEASUREMENTS. 

Such a uniform field as is premised above may be pro- 
duced by a long solenoid surrounding the bar. 

For practical purposes it is sometimes more convenient 
to use a ring of iron instead of a bar. In such a case a 
ring-solenoid is used to produce a field parallel to the 
circumference of the ring. To avoid errors due to vari- 
ations in permeability of the iron when in fields of dif- 
ferent values, the difference between the outer and inner 
radii of the iron ring should be small in comparison with 
either. In such cases the value of &S^ computed along 
the mean circumference, may be taken as the mean 
value for the ring without sensible error. 

133. Magnetic Susceptibility and Permeability. — 
The ratio of the intensity of magnetization S to the 
strength of the field &S is called the magnetic sus- 
ceptibility of the substance. It is denoted by the 

S 
Greek letter k. Thus k= — . It follows that 6B = 

c-jp 

47rc76 

For many reasons it is more convenient to know the 

ratio between Qq and BS^ rather than that between S and 

BS' This ratio is called the magnetic permeability of 

the substance, and is denoted by the Greek letter fx, 

I bus /ji= -^ ' 

&S 

134. Coercive Force When an iron bar or ring 

has been magnetized, it has been noticed that a large 
portion of the magnetization is retained when the mag- 
netizing force has been removed. In a paper by Houston 



MAGNETISM. 281 

and Kennelly ^ the theory has been advanced that the 
residual magnetization is a linear function of the induced 
magnetization. This theory is based on calculations 
made from data giA'en in Swing's Magnetic Inductio7i in 
Iron and Other Metals. Calling BS the intensity of 
field, c73 the resulting magnetic flux, and (S(Bo the residual 
magnetic flux, they found for annealed soft iron wire 
^^ = 0.88 ((£/a — 500). Other samples of iron and steel 
give different values for these constants, but in every 
case the linear relation seems to be true. 

The term coercive force has been loosely used to 
express this tendency to oppose change in the magnetic 
state. Hopkinson uses the term to denote the intensity 
of field which will just restore the iron to an apparently 
neutral condition. 

135. Effect of the Ends of a Bar. — When an iron 
bar is magnetized longitudinally in a uniform field, the 
ends become poles. The effect of these poles is to pro- 
duce a field at all intermediate points of the bar, whose 
tendency is to demagnetize it. If the length of the bar 
is at least fifty times its breadth, it is assumed in practice 
that the bar is equivalent to a very prolate ellipsoid 
whose axes correspond to the length and diameter of the 
bar. The effect of the ends, on this assumption, is given 
by the following equation, d€ — f-fS'— NS\ in which BS 
is the actual field, &S' the original field, and NB the effect 

N 
of the ends. Values for N and — are given in the fol- 

lowing table : " 

^Electrical World, June 1, 1895, p. 631. 

2 Ewing's Mag. Ind. in Iron and Other Metals, p. 32. 



282 



ELECTRICAL ME A S UREMENTS. 



Ratio of 




y 


length to breadth. 


i^. 


4- 


50 


0.01817 


0.001446 


100 


0.00540 


0.000430 


200 


0.00157 


0.000125 


300 


0.00075 


0.000060 


400 


0.00045 


0.000037 


500 


0.00030 


0.000024 



By using a ring instead of a rod this correction is 
avoided. 



136. Mag-netic Inclination or Dip by the Dip- 

NeedleJ — Magnetic inclination or dip is the angle 
which the direction of the earth's magnetic force makes 
with the horizontal. This direction is given by the 
magnetic needle if it is movable without friction about 
an axis at right angles to itself and to the magnetic 
meridian, if, fbst, the axis passes through the centre of 
inertia of the needle, and if, second, its magnetic axis is 
coincident with its geometric axis. These conditions, 
however, will in general fail to be satisfied. As a con- 
sequence the mode of observation described below is 
required. 

The dip circle is a vertical circle movable about a ver- 
tical diameter : the zeros of graduation should be at the 
extremities of a horizontal diameter. The circle should 
be in the plane of the magnetic meridian. A long, slen- 
der compass-needle may be used in making this adjust- 
ment. If the axis of rotation of the circle is verti- 
cal, the bubble of a level will not change on turning the 
circle. The axis of rotation of the needle should pass 

^ Kohlrausch's Physical Measure?ne?its, 3d English Edition, p. 235. Max- 
Avell's Flee, and Mag., Vol. II., p. 113. 



MAGNETISM. 283 

normally through the centre of the graduated circle. 
Four sets of observations are taken, in each of which 
both ends of the needle are read and the mean, called 
the observed angle, is taken: first, (^i, the original posi- 
tion of the needle ; second, \^i , with the reading and the 
movable circle turned 180°; third, (^.., with the needle's 
magnetization reversed and otherwise as in the first set ; 
fourth, -^2 , with the needle and the movable circle turned 
180° again. 

If the apparatus is good and the observations carefully 
made, these four observed angles will be much alike and 
the angle of dip h is expressed as follows : 

h = ^1 + ^1 + ^'^ + ^ 2 
4 * 

If they differ much, it is possible by grinding the side 
of the needle to make <^i and -v/r^ nearly alike and the 
same for (^9 and yfro . Then 

tan S = I Aan 't^+ti + tan ^1+lA. 

If this is not done and ^^ and yfr^ differ considerably, 
we should write 

cot cLi = ^ (cot <^i + cot t/ti), 
cot ci2= - (cot <f>2 + cot i/to) ; 



and finally 



tan ^= ^ (tan a^ + tan ao). 



These expressions are obtained by considering the 
gravitational forces at work resolved into components 
parallel and perpendicular to the magnetic axis. 



284 



ELECTRICAL ME A S UREMENTS. 



Maxwell's method takes into account the relative 
intensities of magnetization in both cases. Calling these 
B^ and D^ and using the same notation as above, 

2(A+i>0 

For fuller explanations the student is referred to 
Kohlrausch and Maxwell as cited above. 



137. Magnetic Inclination by Weber's Earth- 
Inductor. — When a conductor is moved in a magnetic 
field so as to cut lines of force, the time integral of the 
electromotive force generated is equal, in C.G.S. measure, 
to the number of lines cut. If the conductor is a plane 
coil of wire of total area S which makes angles <f)^ and c^o, 
before and after the movement, with the direction of the 
lines of force of a magnetic field of intensity ^, we 
obtain the equation /edt = SEf (sin (/>! — sin </)2). It is 

necessary to 
A Q count (/) from 0° 

—J „ to 360°. 

In Weber's 
earth- inductor 
(Fig. 129) the 
coil of wire Gr is 
usually moimted 
on an axis A in 
its plane. This 
axis is supported 
by a frame F 
mounted on two 
ijL {jt trunnions T, 

F,g. 129. whose axis makes 



TC 




Z JT 



w 



MAGXETISM. 285 

a right angle \\-ith the fost axis. The trunnions are car- 
ried on supports fastened to a platform resting on three 
levelling screws L. For the purposes of this experi- 
ment the axis T should be level and in a magnetic 
east and west line. On the frame are stops Avhich, as 
generally used, limit the angle through which the coil 
may be turned to 180°. Some earth-inductors are 
turned by hand and others are turned by means of 
springs on the removal of a detent. 

The earth-inductor should be joined in series with a 
ballistic galvanometer of long period of oscillation, and, 
if need be, Avith a coil of suitable resistance. On turning 
the coil through 180° an inductive impulse will be felt in 
the galvanometer. The sine of one-half the throw of the 
galvanometer needle will be proportional to the quantity 
of electricit}' passing through the circuit. Three methods 
may be used in producing the deflection. In the first a 
single reversal of the coil gives a single impulse to the 
needle. In the second the coil is reversed each time the 
needle passes through its position of equilibrium, giving 
it successive impulses until no further increase in its 
amplitude is obtained. In the third the coil is reversed 
every second time that the needle reaches its position of 
equilibrium ; as a consequence the impulse causes the 
needle to recoil, it then reaches its maximum amplitude, 
then passes through zero to a smaller amplitude, owing 
to the damping, and on reaching zero recoils, as the coil 
is reversed, to another maximum amplitude in the oppo- 
site direction. This is continued until the arcs of the 
amplitudes reach constant values a and h. 

In the first and second methods the quantities of elec- 
tricity and the time integral of the electromotive force 
are proportional to the sines of one-half the angles 



286 ELECTRICAL MEASUREMENTS. 

of deflection; in the third they are proportional to 
sin - — _- ? For small deflections the scale deflections 

may be taken proportional to the qnantities of electricity 
passing and to the time integral of the electromotive 
force. To reduce deflections to sines of one-half the 
angles, use Table I. in the Appendix. 

Several precautions are to be taken in the use of the 
earth-inductor. As it is assumed that the magnetic field 
is uniform and of constant direction in the neighborhood 
of the coil, the presence of masses of iron and particularly 
of magnets should be avoided. The powerful magnets 
usually in voltmeters and ammeters will noticeably affect 
the lines of force for a distance of several metres. A 
result obtained with an earth-inductor is valuable only in 
the place in which it is obtained ; even in the same room 
considerable variations may be found. In some cases it 
may be due to the iron in the red brick walls and founda- 
tions for piers. Besides the magnetic disturbances within 
our control, there are the daily and yearly variations, of 
which account should be taken in very exact work. 

In the determination of magnetic inclination we may 
make use of a familiar principle, that the direction of a 
vector or directed quantity is completely defined b}^ the 
cosines of the angles included between the line of the 
vector and the three rectangular axes of coordinates 
passing through the point. The component of the vector 
along each of the axes is found b}' multiplying the whole 
vector by the corresponding cosine. In the present 
problem the conditions are chosen so that one component 
is zero, and the vector, the intensity of the earth's field, 

1 Kohlrausch's Phys. Meas., 3d English Edition, p. 351. 



MAGNETISM. 287 

lies parallel to the plane of the other two. The induc- 
tion impulses obtained by reversing the coil are then 
proportional to the vertical and horizontal components 
^V and 8^, and, as a consequence, to the cosines of the 
angles between the lines of force and a plumb line, and 
a horizontal magnetic north and south line respectively. 
These last quantities are also the sine and cosine of the 

inclination, and their ratio, equal to-^^, is the tangent of 

the magnetic inclination or dip. 

First Position^ S^. Place the earth-inductor so that 
the plane of the coil is horizontal and the axis A in the 
magnetic north and south line. An ordinary level and 
a long, slender compass-needle will suffice to secure these 
adjustments. The second condition is desired, as it 
prepares the apparatus for the second position. On 
reversing the coil the number of lines of force cut is 
proportional to the vertical component of the earth's 
field. Observations may be taken in any of the ways 
mentioned above. These observations should be re- 
peated several times and the mean determined. 

Second Position., 98- Turn the frame F through 90°. 
The axis of the trunnions should be in a horizontal 
magnetic east and west line and the axis A vertical. 
The plane of the coil should now be vertical and at 
right angles to the magnetic meridian. The coil should 
be tested for these conditions with the plumb line and 
the compass needle. On reversing the coil the number 
of lines of force cut is proportional to the horizontal 
component of the earth's field. Several sets of observa- 
tions should be taken as in the first position and the 
mean determined. 

Calculation of the Ratio of %^to 96. Strictly speaking. 



288 ELECTRICAL MEASUREMENTS. 

the deflections should be reduced to the sines of one-half 
the angles and their ratio taken. When, however, the 
deflections are not great, we may use in the place of the 
ratio of these sines 

,_iid? 

where di and d^ are the scale deflections in the first and 
second positions, a the distance of the scale from the 
mirror, and 8 the angle of dip. 

138. Determination of the Horizontal Intensity of 
the Earth's Mag-netic Field. — The following method 
of determining ^6 is due to Gauss. It depends on the 
measurement of two quantities, viz., the product and the 
ratio of the horizontal intensity && of the earth's field 
and the magnetic moment 3To of a particular magnet. 

First, to find the Product of 81b and BS- Suppose the 
magnet AB suspended, in the place where &S is to be 
measured, by a bundle of long silk fibres ; a suitable fine 
wire may replace the silk fibres. To ensure freedom from 
torsion a "' dummy " magnet of brass, of weight equal to 
AB., may be hung on the fibres and the torsion head 
turned until the dummy lies in the magnetic meridian. 
Let the magnet so suspended be made to execute tor- 
sional vibrations. Let T be the half period and K be 
the moment of inertia of the magnet, and let 6 be the 
ratio of torsion of the fibres ; then for small amplitudes 



^ '^\l dibdS(i + ey ' ' ' ^^^ 
This value of T should be reduced to the value corre- 



MAGNETISM. 289 

spending to an infinitesimal arc, by Table III. in the 
Appendix. 

The torsional vibrations may be produced by repeat- 
edly presenting first one and then the other pole of a 
strong magnet at a considerable distance from the sus- 
pended magnet. If the change of pole is properly timed, 
the SAving may be greatly multiplied. Conversely if the 
magnet is swinging, it may be brought to rest by pre- 
senting the poles alternately so as to oppose the motion. 
This magnet sliould of course be removed to a great 
distance before the final observations are made. 

By 0, the ratio of torsion, is meant the ratio between 
the restoring forces due to the torsion of the fibres and 
to the action of the magnetic field respectively, when 
the magnet is slightly deflected from the magnetic merid- 
ian. If the tops of the fibres are held by a graduated 
torsion head, and the magnet carries a light mirror, to 
be used in connection with a telescope and scale, the 
ratio of torsion may be readily measured by turning 
the torsion head through an angle a, thereby turning the 
magnet and its mirror through an angle /3. To avoid 
troublesome corrections yS should be so small that it does 
not differ materially from its sine. If equilibrium is 

obtained, 6 — '—-— . 

To find the value of 8Tb • o^ it is necessary to know 
the moment of inertia K of the magnet, equation (1). 
If this cannot be calculated from its dimensions, it may 
be determined experimentally as follows : Take a rftig 
of mass M. and outer and inner radii a^ and a., . Its mo- 
ment of inertia about its axis is \M {a\ + «:>) = K'. 
Place this ring upon the magnet with its centre in 
the line of support. Determine I\, the half period of 



290 



ELECTRICAL ME A S UREMENTS. 



vibration of the system, and correct to an infinitely 
small arc. Then 



r, 






By combining (1) and (2) we obtain 



=p. 



(2) 



.(8) 



Second^ to find the Quotient of STb divided hy BS. 
There are two methods of determining this ratio; in 
both we combine with the earth's magnetic field at 



h' 




a 

B A 



(Figs. IBO, 131), when di3 is being determined, a field 
^due to the magnet AB^ which in both cases has its 
magnetic axis east and west. In the first case the point 
is on the prolongation of the magnetic axis of AB ; in 
the second it is on the perpendicular to the middle point 
offhis axis. In both cases the field g^at (9, due to AB^ 
is directed along the magnetic east and west line. The 
direction of the resultant of BS and c^is indicated by 
N'S'. For convenience in deducing the expressions for 
cf^ more detailed sketches of the positions a (Figs. 130, 



MAGNETISM. 291 

131) are given in Figs. 132, 133. The point is at the 
middle point of ns. 

First 3IetJiod. Let the magnet AB^ used in 
the determination of dlb ' &S^ be placed with 
its positive pole to the east and with its centre 
at a distance r from (position a, Fig. 130). 
Suppose the magnet AB to produce a certain 
deflection of the magnet ns. Reverse AB ; the 
deflection should now be equal and opposite to 
its first value. Next place the magnet at an 
equal distance to the west of 0, and obtain de- 
flections with the positive and negative poles 
respectively directed toward (position 6, Fig. 
130). A pair of deflections equal to the first 
pair should now be obtained. Call the mean of 
these four deflections <^. Repeat these observa- 
tions with AB at a distance r' from (positions 
a' and h\ Fig. 130). Call the mean of the 
deflections in this position ^'. Kohlrausch says,^ 
"In order that the errors of observation may 
have the least possible influence on the result, 

it is best that the ratio of the two distances - 

should equal 1.4;" Gray says- 1.32. The dis- " " 
tance r' should be at least from three to five Rg. isi. 
times the length of AB. 

Combining these two sets of measurements, 

81^ ^1 /-^tan^ — r'Hanc^^ _ ^ , 

Second Method. Let AB be placed in the position a 

1 Phys. Meas., 3d English Edition, p. 243. 

^Absol. Meas. in Elect, and Mag., Vol. 11. , Part I., p. 93. 



■A 



292 



ELECTRICAL MEASUREMENTS. 



(Fig. 131) with its positive pole to the east, and observe 
the deflection of ns ; reverse AB and observe the deflec- 
tion ; repeat in position h ; continne the observations for 
positions a' and h' . Using the same notation as above, 

<97S _ rHan </) — r''' tan ^' 
'as r'-r'- 



&. . (5) 



When the first method is used &8 may be obtained by 
dividing (3) by (4) and extracting the square root. 
Thus 



cip 

ah = 



TT 



V; 



2K(r'~r'') 



(l + e)QT!-T'') {r' tan ^ - r'' tan </> ') 
When the second method is used 






^(r-r'O 



(1 + 6') ( Ti - T') (r' tan ^ -r'^tan (^0 

Proof. Suppose the magnet AB to have its poles of 
strength m^ at a distance l^ apart. Find the force acting 
on a unit pole at a distance r from the centre of AB, 
along its magnetic axis. 

First Method. Let the negative pole of AB (Fig. 132) 




^m 



XwiJs 



Fig. 132. 



be toward the west; the force at due to it will be 



m^ 



('• - ihy 



, directed toward the east. The force due to 



MAGNETISM. 



293 



mi 



^, directed toward the 



the positive pole will be 

west. The total force on unit pole will then be 



cf=mi 



\(:r-¥d' (r+ihY) 



2m,l, 



X 4r^) 



2gj(3r 






1 + 



?T , 3^ 



_A_ 4- etc. 



directed tow^ard the east. Neglecting higher powers than 
the second in the expansion, this may be written 

23m 



8f= 



^(-j> 



(8) 



Second Method. In this method 
let the positive pole of AB (Fig. 
133) be toward the east. Then 
the force on a unit negative pole ^^ 
at a distance r north from the 
middle of AB^ due as before to 
the negative pole of AB, will be 
nil 

(2 , Zf \ directed from B. The 

force due to the positive pole will 
be the same in magnitude, and 
will be directed toward A, For 
convenience in drawing Fig. 133, 
it has been assumed that the poles 
are at the ends of AB. In reality 
they should be further back. Re- 
solving these forces into south and 
east and north and east compo- 




Fig. 133. 



294 ELECTRICAL MEASUREMENTS. 

nents, we fiiid that the north and south components 
annul one another, and the east components produce a 
force on unit negative pole, 

directed toward the east. As before, this may be written 
without sensible error, 

'=?('- 1) (») 

Returning now to the first method, we may suppose a 
short magnet ns (Fig. 132) of length I and pole strength 
m suspended at 0. Call the deflection produced by 
AB </). Then for equilibrium the moments of the two 
couples acting on ns must be equal, or 

&€ml sin (j) = Efml cos <^. 
Therefore 

a^tan(^.= gr=:?^/l + 4,) • • (10) 
When AB i^ in position a' we have 

96un4>'=^^{l+'-^. . . (11) 

Eliminating c from (10) and (11) we obtain 
<97g_l r^tan j>-r^nan<^^ 

In a similar way for the second method we find 
equilibrium of the moments of the two couples when 

a8tan^=f (l-£;), . . (12) 



MAGNETISM. 296 

and g<gtan</.' = ^('l-^,); . . (13) 

which give on eliminating c' 

dlo _ r^ tan (/> — r''^ tan ^' 

Correction for Induced Magnetization. In the meas- 
urement of Silo ' &8 the magnet is suspended in the 
earth's field in such a way that its magnetic moment is 
increased by induction. In very exact work a correc- 
tion should be made for this change. This increase may 
be approximately estimated by the rule that the mag- 
netic moment 81b is increased by — &S per gramme of 

steel. ^ 

Precautions. As the value of d€ is constantly chang- 
ing, and as Sib for a magnet is affected by a tempera- 
ture coefficient, besides being liable to be permanently 
changed by shocks or blows, or by contact with or even 
proximity to other magnets or large masses of iron, it is 
advisable that the whole experiment be performed con- 
secutively. It is unnecessary to add that no iron or 
other magnetic substance near by should be moved 
during the experiment. In general the place in which 
magnetic measurements are made should be free from 
the presence of unnecessary iron. Iron pipes for water, 
gas, or steam, iron window weights, iron telescope bases, 
etc., should be replaced by others made of non-magnetic 
metals. 

139. Measurement of Intensity of Magnetization, 
Magnetic Induction, Permeability, and Susceptibility. 

— When a magnetic substance is undergoing tests with 

1 Kohlrausch's Phys. Meas., 3d English Edition, p. 245. 



296 



ELECTRICAL MEASUREMENTS. 



11000 
10000 
9000 
8000 
7000 
6000 
5000 
4000 

30oa 

onnft 
















a- 








^ 


^ 




^ 


b 








/- 


^ 














/ 


























































1 




















^ — 


^ 








■c 


/ 




^ 












1000 


// 
















I 



















2 4 6 8 10 12 14 JC 



Fig. 134. 



reference to its magnetic qnalities, it is nsnal to deter- 
mine the effect of various magnetic fields in producing 



MAGNETISM. 297 

magnetic indiiction ^3 in the snbstance. From the data 
obtained it is possible to calculate the magnetic per- 
meability yu. of the substance, the intensity of magneti- 
zation S-, and the magnetic susceptibility k. S and k 
are little used, but for many reasons the ideas conveyed 
by these symbols are still useful. 

When a piece of previously unmagnetized iron is 
placed in a magnetic field whose intensity QS is raised 
luiiformly from zero, it is found that the magnetic 
induction increases at first slowly, then by degrees more 
rapidly, until a maximum rate of increase is reached ; 
beyond this point the rate decreases toward a constant 
quantity, which equals the rate of increase of SS as 
a limit, while S' approaches a maximum. If the piece 
of iron has been previously magnetized it may be 
demagnetized by heating to a red heat, or by a process 
of reversals with gradually decreasing field strength. 
Curves «, ^, and c (Fig. 134) represent the relation of 
63 to BS under such circumstances for mild steel, 
wrought iron, and cast iron, respectively. The values 
of the quantities are in C.G.S. units. The data for 
these curves were obtained by experiments on rings, 
using the method of reversals (Art. 145), which does 
not require the demagnetization to be absolutely com- 
plete on starting the tests. 

When the intensity of the field is increased by steps 
from zero to some definite value, decreased from that 
value to zero, increased in the opposite sense to the 
same numerical maximum value as before, again de- 
creased to zero and the cycle repeated, the curve rep- 
resenting the relation of do to 86 after the first 
quarter cycle is similar to that shown in Fig. 135, which 
was obtained from experiments on a cast-iron rinq-. The 



298 



ELECTRICAL ME A S UREMENTS. 



(B 

7000 
6000 
5000 
4000 
3000 
2000 
1000 

1000 
2000 
3000 

4000 
5000 
6000 
7000 


















^-'■""''^ 


?=^ 














y 


^ 


y 
















// 


/ 


















/ 
















1 


/ 


/ 






































\ 








































































































f 

J 


1 
















/ 


/ 
















/ 


/ 


/ 














y^^^ 


/ 


y 














^ 






















86 42034 6 SJC 



Fig. 135. 



IIAGJSfETISM. 299 

fii'st quarter-cycle, not shown in the figure, might have 
been represented by a curve similar to those of Fig. 134. 
It will be noticed that the values of cB corresponding 
to decreasing values of B6 are very much greater than 
those corresponding to the same values of && when in- 
creasing. This magnetic lag in the values of cB, which 
is due to the tendency of the iron to oppose changes 
in its magnetic condition, has received the name of 
magnetic hj'steresis. As curves of cyclical magnetiza- 
tion show hysteresis, they are commonly called hysteresis 
curves. The amount of energy expended in each cubic 
centimetre of iron per cycle because of hysteresis is 

TF= 1 fd&dm = Area of Curve 

•iTT 4:17 

when the curve is plotted to scale. 

Four well-known methods have been used to determine 
the relation of &6 to cB, j\ f^-, and /c ; the optical 
method, used by du Bois,^ depending upon the phenom- 
enon discovered by Dr. Kerr,- that when plane polarized 
light is reflected by a magnet pole, the plane of polar- 
ization is turned through an angle depending upon the 
intensity of magnetization ; the magnetometric method ; 
the tractional method ; and, finally, the ballistic method. 
Of these the fust will not be considered here further, 
inasmuch as it requires a practised experimenter to 
obtain good results. 

140. The Magnetometric Method. — This method is 
applicable to open magnetic circuits only. The theory 
of the method is similar to that used in the determination 
of the horizontal component of the earth's field, which 

1 Phil. Mag., March, 1890, April, 1890. 

2 B. A. I^eporf, 1876, p. 40 ; Phil. Mag., May, 1877. 



300 ELECTRICAL MEASUREMENTS. 

ill this section will be designated by &Se • The magnet- 
ometer consists essentially of a small magnet suspended 
by a fibre of little torsion. In order that the deflections 
may be read, the magnet carries with it a light mirror. 
In practice the fibre may be attached to a mirror on the 
back of which several small magnets are cemented. If 

the larger magnet used in the determination of — "^ 

replaced by that on which the experiment is to be made, 
a single observation will give dJb if - in equation (8) 
be neglected ; for 

9 QTT' -J 

BSe tan <h = — ^ , and 8Tb = - r® &Se tan 6, 

r' 2 

for the first method ; or 

d€e tan (/> = ^^, and 87b = r^ B8e tan <^, 

r" 

for the second method. 

If V be the volume of the magnet and a solenoidal 

81b 
magnetization be assumed, then S = — • 

/^ 
It is found, however, in practice, that —, should not be 

neglected. Furthermore, the position of the poles is not 

at the ends and J is not uniform or solenoidal. In the 

case of a bar in the form of a very prolate ellipsoid of 

revolution, of minor axis a and length Z, the distance 

21 
between the poles is ~, and the following formula is 
o 

obtained : 

''^^g^«tan<^ 



i^-y 



// = 7^ for the first method, 



MAGNETISM. 301 



6(.^4y 



BSe tan (/) 



and S = ~i for the second, method. 

The last formula is frequently applied to long cylin- 
drical bars and leads to little error. 

One-Pole Method. A better method is to place the 
bar under test in a vertical position and east or west from 
the magnetometer. When placed in this position it is 
found that the bar is affected by the vertical component 
of the earth's field unless this component is compensated 
by a solenoid about the bar. The current through the 
solenoid will, however, affect the magnetometer, unless 
the horizontal component of the field produced at the 
magnetometer needle b}^ the solenoid, with the bar 
removed, is compensated by another solenoid placed 
with its axis horizontal and in an east and west line 
passing through the magnetometer needle. The same 
current should pass through both solenoids, and the 
relative distances should be arranged so as to annul the 
effect at the magnetometer. The compensation is then 
assured with all currents. The current should also pass 
through an ammeter and an adjustable resistance to 
insure permanent compensation of %^ at the bar. The 
magnetizing solenoid also should surround the bar. 

The height of the bar should be adjusted until, witli 
a certain magnetization, a maximum effect is obtained on 
the magnetometer. It is then assumed that one pole is 
directly behind the magnetometer. If the bar is long 
the effect of the lower pole is very slight. Assuming 
that the poles are at equal distances from the ends, the 
upper one at a horizontal distance n from the magnet- 
ometer needle and the lower one at a distance r, along 



302 ELECTRICAL MEASUBEMENTS. 

a line of inclination ^, and calling the distance between 
the poles I and the cross-section S^ we have 



a^etanc/) =SS -- 



1 cos 6> ) SS 



or 



&Se tan ^rj 



-CiJ 



S\ 1 



(?j 



If BSe is i^ot known, it maj^ be found by comparison 
with the intensity of the field produced at the magnet- 
ometer needle by the second compensating horizontal 
solenoid. 

When the positions of the two compensating solenoids 
and the bar have been adjusted, the next step to be taken 
is to demagnetize the bar by reversals. For this there 
should be introduced into the circuit of the magnetizing 
solenoid a resistance adjustable by small steps from zero 
to its full value, and a commutator to reverse the current. 
The adjustable resistance should be cut down until the 
magnetization of the bar is as great as any value reached 
since its last demagnetization. The direction of the 
current should be continually and rapidly reversed while 
the adjustable resistance is increased gradually to its 
highest value, and finally the circuit should be broken. 
A liquid resistance, such as zinc sulphate solution 
between zinc plates, whose distance apart may be varied, 
makes a satisfactory adjustable resistance. If the magnet- 
ometer does not return to its zero reading, the current 
through the compensating solenoids should be changed 
until it does. As feeble magnetic forces are slow in 
acting, it is necessary to allow some time for this adjust- 
ment. 

This method is very valuable for the investigation of 



MAGNETISM. 



303 



the effects of weak fields on 6B, S-, /^^ and k. For such 
work the ballistic method is quite unsatisfactory, owing 
to the creeping up of the magnetization. 

Example.' 

Test of a piece of wrought-iron wire by the magnetometric 
method. Cross-section of wire, 0.004658 sq. cm. ; length of wire, 
30.05 cms. ; ^^ equalled 0.299 C.G.S. unit. 

Distance of millimetre scale, 1 metre; ri = 10 cms., ^2== 31 



cms. Whence 



(sy-- 



0335. 



Deflection of one scale part corresponds to tan = 0.0005. 
Value of '/ per scale division = 



0.299 X 0.00 05 X 100 
0.004658 X 0.9665 
The magnetizing coil contained 69 turns per cm. 



= 3.32. 



Magnetizing force per ampere = 



4-. 69. 
10 



86.7. 



Magnetizing 












force &8 


Magnetometer 




cf 


m = 


SB 


Selenoid 


Corrected 


readings. 


c) 


&s 


Arrr7+ &6 


'^96 


alone. 


for ends. 












0.32 


0.32 


1 


3 


9 


40 


120 


0.85 


0.84 


4 


13 


15 


170 


200 


1.38 


1.37 


10 


33 


24 


420 


310 


2.18 


2.14 


28 


93 


43 


1170 


550 


2.80 


2.67 


89 


295 


110 


3710 


1390 


3.50 


3.24 


175 


581 


179 


7300 


2250 


4.21 


3.89 


239 


793 


204 


9970 


2560 


4.92 


4.50 


279 


926 


206 


11640 


2590 


5.63 


5.17 


304 


1009 


195 


12680 


2450 


6.69 


6.20 


.327 


1086 


175 


13640 


2200 


8.46 


7.94 


348 


1155 


145 


14510 


1830 


10.23 


9.79 


359 


1192 


122 


14980 


1530 


12.11 


11.57 


365 


1212 


105 


15230 


1320 


15.61 


15.06 


373 


1238 


82 


15570 


1030 


20.32 


19.76 


378 


1255 


64 


15780 


800 


22.27 


21.70 


380 


1262 


58 


15870 


730 



141. The Tractional Method. — If a longitudinally 
magnetized bar be cut orthogonally in two, and the parts 



2 Ewing's Mag. Ind. in Iron, p. 49. 



304 ELECTBICAL MEASUREMENTS. 

be separated an infinitesimal distance, both end surfaces 
will show equal intensities of magnetization j. Call 
the area of each end surface S. The attraction of one 
surface on the other will be ^ttcT'S^ provided the field 
BS about the bar be negligible. If BS is not negligible 
and is due to an outside cause, — for example, a magnetiz- 
ing solenoid not attached to the magnet, — we must add to 
the above a force BScJS. If the solenoid is in two parts 

closely wound about the bar, which separate with the 

0/52 a 
parts of the bar, we must add a third term — — for the 

oTT 

mutual attraction of the two parts of the solenoid, which 
is assumed to be of the same cross-section as the bar. 
These forces are in djnies ; to reduce to grammes they 
must be divided by 980. Reducing to a common 
denominator and substituting the value of 6(3, we obtain 
for F in grammes under the three conditions, 

F=/-(1Q^^S^) = ^ («) 

^Trg Sttc/ 

F=J- (IQtt'-J' + SttJ&S) = (^JZ^O^. . (5) 
F=^ (IGtt^ J' + SttcI&S + &&') = —. . (0 
Also m = 'xl^IEl=156.9^/l. . (aande) 

m = ^^+Sf6'- (b) 

It is evident from the above equations that if (^ and 
6B are not uniform over the whole cross-section of the 



MAGNETISM. 305 

magnet, the result obtained will be the square root of 
the mean square, and not the simple mean. The square 
root of the mean square is always greater than the 
mean. It therefore follows that the value here obtained 
may be slightly larger than that obtained by other 
methods. Exactly such results were obtained from ex- 
periments made with a horse-shoe magnet (Fig. 141). 
The upper curve of Fig. 142 represents the relation of 
6B and ^, with the values of 6B calculated from the 
force necessary to detach the armature. The lower 
curve was obtained by the ballistic method (Art. 145), 
the exploring coil being in the position marked 2. For 
large values of BS the value of cB tends to become 
uniform over the whole cross-section, and the curves 
approach each other. 

142. The Divided Ring. — Let a divided ring (Fig. 
136) of cross-section S 'sq. cms. be uniformly wound 
with a magnetizing coil of n turns per 
cm., measured along the mean circum- 
ference ; and let the coil be traversed 
by a current of / units in C.G.S. 
measure. Then the value of the mag- 
netizing force &S will be ^irnl. At- 
tach the hook on the lower half C 
to the bottom of a frame, from the top 
of which the upper half C is supported 
by means of a spring balance, hooking 
into the eye on C\ and a turn-buckle to Fig. i36. 

increase the tension. Care should be 
used in setting up the apparatus so that the line of pull 
may pass vertically through the centre of the ring and 
normally to the plane of separation of the two halves. 




306 ELECTRICAL MEASUREMENTS. 

Either the balance should be adjusted to zero when 
supporting (7^, or else the weight of C^ should, be sub- 
tracted from the observed forces. In calculating 6^, 
the value of S should be doubled so as to include 
both surfaces of contact. 

Shelford Bidwell made use of a ring made from a 
very soft charcoal iron rod. After welding the joint, 
the ring was turned down in a lathe to a uniform trans- 
verse circular cross-section of 0.482 cm. diameter. The 
outer radius of the ring was 4 cms. and the mean radius 
3.76 cms. After the ring was sawed in two, brass col- 
lars were fastened to the ends of one half to hold the 
other half in position, thus insuring freedom from 
lateral displacement. Both halves were uniformly 
wound with ten layers of insulated wire 0.07 cm. in 
diameter. After each turn was in place the radial gaps 
were filled with paraffin. The half with the collar had 
980 turns, part of which were on the collar, and the 
other half had 949 turns. When the two halves were 
together, the ring appeared to be uniformly wound 
without break. The value of &S was carried to 585, 
when the weight supported by the ring reached 15,905 
grammes. 

143. The Divided Rod. — A rod is a more convenient 
form for testing than the ring, since it does not need to 
be bent, welded, and turned true. The apparatus used by 
Bosanquet^ for testing rods is shown in Fig. 137. The 
rod is divided into halves c, c\ which meet in carefully 
faced surfaces. Each half has a closely wound solenoid 
B about it. From the bottom of <? a scale pan is sus- 

1 Phil. Mag., Vol. XXII., 1886, p. 535. 



MAGNETISM. 



307 



penclecl. The scale pan, the lower half of the rod, and 
its solenoid are counterbalanced by a lever not shown in 
the figure, so that when the 
pan is empty there is no sep- 
arating force at the junction. 
An exploring coil i>, connected 
with a ballistic galvanometer, 
surrounds the upper end of c. 
When c and & separate, B is 

quickly withdrawn from the - 

field by a spring, thus giving 
an independent method of 
measuring cB- Bosanquet 
found a close agreement be- 
tween the values of 6B cal- 
culated by the two methods for 
large values of BS- For small 
values the agreement was not 
good. He used two cylindrical 
iron rods 20 cms. long and 

to Fig. Vii. 

0.526 cm. diameter, each wound 

with 1,096 turns of wire. The maximum weight sup 

ported was 20,414 grammes. 



l2) 



144. Thompson's Permeameter. — Bosanquet's 
divided ring method has the disadvantage of having poles 
at the ends of the divided bar. Allowance must be made 
for these in computing BS. Thompson has avoided this 
difficulty by slotting out a rectangular block of iron A 
(Fig. 138) to receive a magnetizing solenoid ^, through 
which a coaxial brass tube passes. The sample to be 
tested is turned into a cylinder just fitting the brass 
tube, and is inserted from the top. The lower end of 



308 



ELECTRICAL MEASUREMENTS. 



e is carefully surfaced and rests against a part of the 
yoke, which is also carefully surfaced. The lines of 
force are assumed to go through 
the iron only. ^ is calculated as 
before from the pull which will over- 
come the magnetic attraction at the 
lower end of the rod. The attrac- 
tion at the upper end of the rod is 
at right angles to the rod and has no 
effect. Tliompson gives the for- 
mula 




6g == 1317 VPounds -^ sq. in. + &8 = 
156.9 VGrammes H- sq. cms. + &S 

to express the relation between 68, 
&S^ and F^ wliich is somewhat differ- 
ent from formula (5), Art. 141. Er- 
rors of observation will, however, 
cause larger differences than that between the for- 
mulas. 



Fig. 138. 



145. The Ballistic Method. — This method in its 
present form is due to Rowland.^ It depends upon the 
principle that when the flux of magnetic induction 
through a coil S (Fig. 139) of Ux turns is changed 
by a quantity iV, the time integral of the electromotive 
force generated in the coil is n^N. If the coil S be in 
a circuit of resistance r, including a ballistic galva- 
nometer G- of long period, the quantity of electricity 



passing through the galvanometer will be 
these quantities are in C.G.S. units. 



ri^N 



All of 



Phil. Mag., Vol. XLVI., 1873, p. 151. 



MAGNETISM. 



309 



If the same circiiit includes, as part of r, an eartli- 
inclnctor EI of total 
area A lying horizon- 
tally in a place where 
the vertical component 
S^ of the earth's mag- 
netic field is known, 
the constant h of the 
galvanometer may be 
determined by a simple 
reversal of the coil. 
Let cZi be the deflection 



(corrected to 



. 1 

sm - 

2 

angle) corresponding 
to the quantity of 
electricity Q passing 
through the galvanometer; then by Art. 137 




Fig. 139. 



Q = djc 



2^9 



, and k = 



div 



Let d2 be the corrected deflection due to the change 



iV^in the flux through S ; then t^s^ 
follows that 



n,N 



, from which it 



N=: 



2AVd2 



Uidi 



(1) 



There are several objections to -the use of the earth- 
inductor. In the first place, an error may be made in 
determining A ; next, an error may be introduced by a 
change in S^, due to any one of many causes ; and, 
thirdly, a considerable error may be introduced because 



310 ELECTBICAL MEASUREMENTS. 

of the large number of observations necessary in deter- 
mining S^. For these reasons it is better to determine 
k by means of a standard cell and a standard condenser. 
Let JE be the electromotive force of the standard cell, 
and C the capacity of the condenser. Connect the 
apparatus as in Fig. 88, p. 188, and charge the condenser 
and discharge it through the galvanometer. Let d^ be 
the deflection ; then dji: — CE., and 

N=10Q ^2<^-^^ ^ .^^ 

d^sUi 

where C is in microfarads, E in volts, and r in ohms. 
The resistance r is the resistance of the entire circuit as 
connected for the determination of iV! 

As the damping of a ballistic galvanometer may be 
appreciably different on open and on closed circuits, it 
is ad^dsable to have the discharge key double so as to 
close the galvanometer through an external resistance 
equal to that of the working circuit immediately after 
the discharge of the condenser. 

For this method it is better to use the iron in the form 
of a ring. It should be wound uniformly all the way 
around with a primar}- coil P, of ^2 turns per cm. meas- 
ured along the mean circumference. P should be in 
series with an ammeter, not shown in Fig. 139, a resist- 
ance Hi adjustable by small gradations, and a storage 
battery SB^ through a commutator C. If a current of 
I C.G.S. units flows through this circuit, the correspond- 
ing value of &6 will be -^irn.jL 

To find A6B, the change in the value of cB, the 
change of flux in the iron should be divided by the 
cross-section A' of the iron. To be exact, the part N' 
included between the iron and the secondary coil S^ 



MAGNETIS3L 311 

which is wound outside of the primary, should be 
subti'acted from J^. But this correction is generally 
negligible. 

Practice of the Method. When ready to begin the 
experiment, the ring if previously used should be 
demagnetized by reversals, beginning with the highest 
value of d€ employed before. The resistance in Ri 
should be gradually increased after each reversal of the 
commutator until its highest value is reached, and the 
circuit should then be opened. 

To obtain a simple magnetization curve by reversals, 
the value of Hi should be adjusted to give the lowest 
value of &8^ the circuit closed, and the value of the 
current observed. The commutator should then be 
reversed and the deflection of the galvanometer noted. 
As the flux through S is only one-half the change in 
the flux, the value of cB should be calculated from one- 
half the deflection. To obtain the residual value of 
6B, the circuit should be broken and the deflection 
again noted. This residual value is proportional to the 
difference between this deflection and half the previous 
one. Ml should now be decreased for the next higher 
value of &€•, and the observations repeated, and so on. 
The values of &6 when plotted with the corresponding 
values of S2 will give the curves of temporary and 
residual magnetization. 

To obtain a cyclical magnetization or hysteresis curve, 
the ring should be demagnetized as above. Then, 
adjusting the value of Hi for the lowest value of Btj 
desired, the circuit should be closed, the deflection of 
the galvanometer noted, and the ammeter read. The 
resistance Ei should now be decreased abruptly by suit- 



312 ELECTRICAL MEASUREMENTS. 

able steps and the corresponding deflections noted. The 
corresponding values of 6B are proportional to the sum- 
mations of the deflections from the beginning. When 
the highest value of EiS desired has been reached, the 
resistance Ri is increased by suitable steps, and B8 
reduced until on breaking the circuit SiS is zero again. 
The commutator is now reversed and BS is carried 
to corresponding values in the opposite sense, and 
so on. After the fu^st quarter-cycle the values of &8 
and 6B repeat themselves, and the resulting curve 
is called a cyclical magnetization or hysteresis curve. 
The first quarter plots as a simple magnetization 
curve. 

There should be little difference between the outer 
and inner radii of the iron ring used in this experiment. 
If a bar is used it should be at least forty diameters in 
length and the magnetizing solenoid should cover almost 
the whole length. The secondary coil should be at the 
centre. To avoid the effects oi the earth's field the axis 
of the ring should be along the lines of force, but it is 
sufficient if the ring is horizontal and the axis of the 
secondary coil is east and west. In the case of a bar, 
the axis should be east and west. 

For convenience in bringing the galvanometer needle 
to rest, a small coil, through which the magnet ns may 
be thrust, is included in the circuit of G. Every move- 
ment of n8 produces an induced current which may be 
so timed as to check the swing of the needle. A sole- 
noid near the galvanometer in circuit with a single cell 
and a key within reach of the observer may serve the 
same purpose. 

The great fault in the ballistic method as applied to 
rings is that it takes no account of the gradual changes 



MAGNETIS2L 313 

in maguetization — the so-called creeping up — which 
follow any sudden change in Btj. Hopkinson's bar and 
yoke method, described in the following article, is to a 
large extent free from this defect. 

Example. 

The Ballistic Method applied to a Cast-iron Ring. 

Total area of earth-inductor^, 48,600 sq. cms. 
Vertical component of the earth's field S^, 0.54. 
Corrected deflection of the galvanometer for one turn of earth- 
coil, di, lb. 

Xumber of turns in 5, ?2i 20. 

Xumber of turns in P 273. 

Mean length of magnetic circuit . . . . 39.82 cms. 

Number of turns per cm., ?Z2 6.86. 

Cross-section of ring, ^' 5.94 sq. cms. 

No allowance was made for N' . Hence 

&8 = ^.-n^I=^ 86.2/C.G.S. units. 

.V 2AQpd^ 

AB = ~= :=5.89 d2. 

A' A'nidi 

B = d.S9^d2. 

The ring had been previously used, and had not been com- 
pletely demagnetized before the beginning of the test, and as a 
consequence the values of 68 for the first quarter-cycle do not 
represent changes from a neutral condition. One-half the 
numerical difference between the extreme observed values of 
tB will, however, give the real initial value of cB. Applying 
this as a correction, the real values of 68 are obtained. The 
correction in this instance was 1,758. 



314 



ELECTRICAL ME A S UREMENTS. 



/(C.G.S.) 


&s 


d2 


2(f2 


6B 


CJQ corrected. 


-f 0.058 


+ 5.00 


+ 75.5 


+ 75.5 


+ 445 


— 1313 


0.079 


6.81 


40 


115.5 


680 


1078 


0.126 


10.86 


363 


478.5 


2818 


+ 1060 


0.148 


12.76 


165.5 


644 


3793 


2035 


0.176 


15.] 7 


147 


791 


4659 


2901 


0.217 


18.71 


150.5 


941.5 


5542 


3784 


0.269 


23.19 


138 


1079.5 


6358 


4600 


0.356 


30.69 


190.5 


1270 


7480 


5722 


0.490 


42.24 


105 


1375 


8099 


6341 


0.669 


57.67 


142 


1517 


8935 


7177 


1.117 


96.29 


215.5 


1732.5 


10201 


8443 


0.490 


42.24 


— 210 


1522.5 


8968 


7210 


0.307 


26.46 


96.5 


1426 


8399 


6641 


0.127 


10.95 


153 


1273 


7498 


5740 


■ 0.056 


4.83 


95 


1178 


6938 


5180 


0.000 


0.00 


107 


1071 


6308 


4550 


— 0.040 


— 3.45 


127 


944 


5560 


3802 


0.058 


5.00 


93 


851 


5011 


3253 


0.077 


6.64 


199 


652 


3840 


2082 


0.107 


9.22 


395.5 


256.5 


1511 


— 247 


0.122 


10.42 


152.5 


104 


612 


1146 


0.142 


12.24 


169 


— 65 


— 383 


2141 


0.172 


14.83 


156.5 


221.5 


1305 


3063 


0.209 


18.02 


155 


376.5 


2218 


3976 


0.267 


23.02 


137.5 


514 


3027 


4785 


0.390 


33.62 


178 


692 


4076 


5834 


0.508 


43.79 


106 


798 


4700 


6458 


0.694 


59.82 


134 


932 


5489 


7247 


1.105 


95.25 


202.5 


1134.5 


6682 


8440 


0.456 


39.31 


+ 220.5 


914 


5383 


7141 


0.304 


26.20 


92 


822 


4842 


6600 


0.127 


10.95 


145.5 


676.5 


3985 


5743 


0.056 


4.83 


91.5 


585 


3446 


5204 


0.000 


0.00 


104 


481 


2833 


4591 


+ 0.057 


+ 4.91 


188 


293 


1726 


3484 


0.0572 


4.93 


33 


260 


1531 


3289 


0.070 


6.03 


98.5 


161.5 


951 


2709 


0.086 


7.41 


179 


+ 17.5 


+ 103 


1655 


0.095 


8.19 


130 


147.5 


869 


889 


0.107 


9.22 


140.5 


288 


1696 


62 


0.123 


10.60 


168.5 


456.5 


2689 


+ 931 


0.143 


12.33 


158.5 


615 


3622 


1864 


0.172 


14.83 


146.5 


761.5 


4485 


2727 


0.208 


17.93 


156.5 


918 


5407 


3649 


0.264 


22.76 


144.5 


1062.5 


6258 


4500 


0.384 


33.10 


220 


1282.5 


7554 


5796 


0.491 


42.33 


110.5 


1393 


8205 


6447 


0.697 


60.08 


138 


1531 


9017 


7259 


1.088 


93.79 


200 


1731 


10196 


8438 



146. Hopkinson's Bar and Yoke Method. — This 
method makes use of a piece of apparatus very much 
like Thompson's permeameter. The method, however, 
is a ballistic one. The bar to be tested is divided into 



Fig. 140. 



MAGNETISM. 315 

two pieces c and c^ (Fig. 140), the former movable in 
the direction of its length, and the latter fixed dur- 
ing the test. The 

abutting ends of 
the two pieces must 
be carefully sur- 
faced and in close 
contact. The lat- 
eral surfaces of (7, 
c^ should be in good contact with A, which is made 
of very soft iron. The solenoid BB is divided in 
halves, and between them is a test coil I) wound 
on an ivory ring. This test coil is connected in cir- 
cuit with a ballistic galvanometer. When the part c 
of the bar is abruptly drawn out a short distance by 
means of the handle, a spring throws the test coil out 
from the yoke, thereby making it cut the whole flux of 
induction present when the handle was pulled. The 
deflection of the galvanometer measures the flux. For 
cyclical magnetization curves the parts of the bar are 
not separated, and the apparatus acts in all essential 
respects like the ring of the previous section. 

Because of the small value of the magnetic reluc- 
tance in ^, it is assumed that the equivalent length 
of the magnetic circuit is the length of the slot in 
A. It is also assumed that there is no leakage of 
magnetic induction from the bar. Although these 
conditions are not exactly fulfilled, and although the 
reluctance of the joints is not insensible, yet for prac- 
tical commercial purposes the method is sufficiently 
exact. 



316 



ELECTRICAL ME A S UBEMENTS. 



3 A 



5 C D 




Fig. 141. 



147. Comparison of the Values of H^ by the Bal- 
listic and the Tractional Methods. — The electro- 
magnet, shown on a scale of one-sixth in Fio-. 141, 

was designed to test the 
laws of traction, both with 
the armature in contact and 
when separated from the 
pole piece at various dis- 
tances. In the course of 
the experiments made upon 
it (9a was measured by both 
the ballistic and tractional 
methods. When the arma- 
ture was in contact the re- 
sults were exactly what 
might have been anticipated 
from theory. With low^ val- 
ues of ciS tliere should be a considerable difference be- 
tween the values of 8h on the inner and the outer sides 
of the magnetic circuit ; and as a consequence the mean 
value of ^73, determined by the ballistic method, should 
be less than the square root of the mean square value, 
determined by traction. For higher values of EfS the 
value of IB tends to become uniform over the whole 
cross-section, and as a consequence the mean value and 
the square root of the mean square value tend to be- 
come equal, and the results b}- the two methods should 
agree closely. Fig. 142 gives t)^>&S curves calculated 
by both methods, the upper one by the tractional and 
the lower by the ballistic method. 

The traction ^^-as applied at the middle of the arma- 
ture by means of a spring dynamometer and a lever 
combined. ^\.s the spring d3-namometer read no higher 



MAGNETISM. 



31' 



than 50 kilos., it was necessary to have recourse to a 
lever in addition. The method of operating was to 
place large weights in the pan attached to the free end 



(B 


















18000 










^^ 




_^_ 


9 


16000 

14000 
12000 








^ 


"y^ 














/ 














/ 
















10000 
8000 
6000 
4000 

2000 


/ 
















/ 
















/ 
















/ 
















1 

















20 



40 



60 80 100 

Fig. 142. 



120 140 Q^ 



of the lever until the proper value was nearly reached; 
then by means of a turn buckle, the pull was increased 
by drawing up the dynamometer until the armature 
was detached. The value of Qo was calculated from 



818 ELECTRICAL MEASUREMENTS. 

the formula, cB = 156.9^/— -, where 1" was the pull 

in grammes and jS the cross-section in sq. cms. In this 
particular case S was 11.34. Therefore 

6B = 32.94\/1\ 

The galvanometer constant was determined b}- means 
of a standard Carhart-Clark cell, whose electromotive 
force was 1.432 volts at 29° C, and an Elliott condenser 
of 0.5065 mf. capacity. The condenser when charged 
by the cell and discharged through the galvanometer 
gave a corrected deflection do of 39.4. When the 
galvanometer was connected with test coil 2 of one 
turn, rii = 1, the resistance of the circuit was 6660 
ohms. Calling do the deflection caused by reversing 
the magnetic flux through coil 2, we have by Art. 
145, 

m = ioof^=54.id,. 

2niAf«3 

The total number of turns in ABCD was 3464, and 
the equivalent length of the magnetic circuit was com- 
puted to be 83 cms., making the number of turns per 

cm. n.2 equal to -^^. Consequently, calling the cur- 

rent J, 

B8 = ^7rnl= 525 I C.G.S. units. 

The following table gives the results by both 
methods : 



MAGNETISM. 



319 



Tractional Method. 


Ballistic Method. 


Current. 


dS 


Grrammes. 


m 


Current. 


96 


Deflec- 
tion. 


a. 


0050 

0.110 

0.160 
0.200 
0.257 


26.2 

57.7 

83.9 
105.0 
134.9 


196000 

243000 

263000 
270000 
273000 


14580 

16240 

16890 
17120 
17210 


0.026 

0.050 

0.076 

0.1012 

0.133 

0.170 

0.210 

0.270 


13.6 
26.2 
39.9 
53.1 
69.8 
89.2 
110.0 
141.7 


17.4 

24.45 

27.6 

28.9 

30.2 

30.8 

31.6 

32.2 


9410 
13230 
14930 
15630 
16340 
16660 
17090 
17420 



148. Magnetic Leakage. — To determine the value 
of the magnetic flux with various numbers of ampere- 
turns and in various parts of the magnetic circuit, six 
test coils were wound at points designated by the num- 
bers 1 to 6 (Fig. 141). Coil 1 could be moved to the 
position la. With the armature in contact with the 
pole pieces, the value of the flux through the several 
test coils was determined for various numbers of 
ampere-turns in J., B, C\ and D, which were always in 
series. 

With the armature in contact with the pole pieces, 
the maximum flux was through coil 3, the flux decreas- 
ing through the other coils in the order 4, 2, 1, 6. The 
flux through 5 was not measured. When the armature 
was separated from the poles by a distance of 0.32 cm., 
and a smaller number of ampere-turns than 6300 was 
used, the order was 4, 6, 3, 1, 2; above 6300 ampere- 
turns 6 and 3 exchanged places. When the armature 
was in contact with the poles there was leakage from 
its ends. This was shown by the fact that the deflec- 
tion produced by coil 1 was reversed in direction when 
placed at la. With the armature at 0.32 cm. from the 
poles, however, the flux through the ends was added to 



320 ELECTRICAL MEASUREMENTS. 

that through the middle of the armature, the deflec- 
tions at 1 and la being of the same sign. At 1.27 cms. 
coil 5 showed the greatest flux ; coils 4 and 6 came next, 
6 leading slightly up to 5000 ampere-turns and beyond 
that coil 4; the others followed in the order 3, 1, 2. 
The same relative order was maintained when the arma- 
ture was removed from the poles a distance of 6.35 
cms. In every case coil 6 was traversed by a greater 
flux than coil 4 for the smaller numbers of ampere- 
turns, the reverse being true for the larger num- 
bers. As the distance of the armature from the poles 
increased, there was an increase in the number of 
ampere-turns at which the exchange of relative values 
between 4 and 6 took place. This exchange is explained 
by the increased reluctance of the iron portion of the 
magnetic circuit with the higher values of cB. 



APPENDIX A. 



321 



APPENDIX A. 



TABLE I. 

Reduction of Deflections observed with Mirror and Scale (Art. 28 \ 
d = deflection reckoned from the point of rest. 
a = distance bet^Yeen mirror and scale. 

a 

n 
The values of 6, tan ^, sin (9, 2 sin - are obtained by 

multiplying ^ l)y the factor corresponding to the value of 

S in the table. This factor is equal to unity diminished 
by the value of the expression standing at the head of 
the column. 





e 


tan0 


sine 


2 8in^ 


s 


252 


162 


l62 






2 


4 


8 


l_l62 
32 


0.01 


0.99997 


0.99998 


0.99996 


0.99997 


0.02 


0.99987 


0.99990 


0.99985 


0.99986 


0.03 


0.99970 


0.99978 


0.99966 


0.99969 


0.04 


0.99947 


0.99960 


0.99940 


0.99945 


0.05 


0.99917 


0.99938 


0.99906 


0.99914 


0.06 


0.99880 


0.99^0 


0.99865 


0.99876 


0.07 


0.99837 


0.99878 


0.99816 


0.99832 


0.08 


0.99787 


0.99840 


0.99760 


0.99780 


0.09 


0.99730 


0.99798 


0.99696 


0,99722 


0.10 


0.99667 


0.99750 


0.99625 


0.99656 




Ia2_^l64 


A 62 ^1^4 


^62+ M.64 
8 ^128 


Il62+i3l5. 




3 ^5 


4 ^8 


32 2048 


0.11 


0.99600 


0.99699 


0.99550 


0.99587 


0.12 


0.99525 


0.99643 


0.99465 


0.99508 


0.13 


0.99443 


0.99582 


0.99372 


0.99423 


0.14 


0.99355 


0.99515 


0.99272 


0.99331 


0.15 


0.99260 


0.99444 


0.99166 


0.99233 


0.16 


0.99160 


0.99368 


0.99054 


0.99129 


0.17 


0.99054 


0.99288 


0.98935 


0.99020 


0.18 


0.98941 


0.99203 


0.98809 


0.98905 


0.19 


0.98823 


0.99n4 


0.9S677 


0.9S775 


0.20 


0.98700 


0.99020 


0.98539 


0.9S659 



B22 



ELECTRICAL MEASUREMENTS. 



TABLE II. 

Reflecting Galvanometer Scale Errors. 

(These corrections are to be subtracted from the observed deflections. Art. 28.) 



Distance 
a. 


1000 


1050 


1100 


1160 


1200 


1250 


1300 


1350 


1400 


1450 


1500 


50 
60 
70 


0.05 
0.05 
0.1 


0.05 
0.05 
0.1 


0.05 
0.05 
0.05 


0, 

0.05 

0.05 


0. 

0.05 

0.05 


0. 

0.05 

0.05 


0. 

0.05 

0.05 


0. 

0.05 

0.05 


0. 

0.05 

0.05 


0. 

0.05 

0.05 


0. 
0. 
0.05 


80 

90 

100 


0.15 

0.2 

0.25 


0.1 

0.15 

0.2 


0.1 

0.15 

0.2 


0.1 

0.15 

0.2 


0.1 

0.15 

0.2 


0.1 
0.1 
0.15 


0.1 
0.1 
0.15 


0.05 

0.1 

0.15 


0.05 

0.1 

0.15 


0.05 

0.1 

0.1 


0.05 

0.1 

0.1 


110 
120 
130 


0.35 
0.45 
0.55 


0.3 
0.4 
0.5 


0.25 
0.35 
0.45 


0.25 

0.3 

0.4 


0.25 
0.3 
0.4 


0.2 
0.3 
0.35 


0.2 

0.25 

0.3 


0.2 

0.25 

0.3 


0.2 

0.25 

0.3 


0.15 

0.2 

0.25 


0.15 

0.2 

0.25 


140 
150 
160 


0.7 

0.85 

1.0 


0.65 

0.8 

0.95 


0.55 

0.7 

0.85 


0.5 

0.65 
0.8 


0.5 
0.6 
OJ 


0.45 
0.55 
0.65 


0.4 
0.5 
0.6 


0.35 
0.45 
0.55 


0.35 
0.4 
0.5 


0.35 
0.4 
0.5 


0.3 

0.35 

0.45 


170 
180 
190 


1.2 
1.4 
1.65 


1.1 
1.3 
1.55 


1.0 
1.2 
1.4 


0.95 

1.1 

1.3 


0.85 

1.0 

1.2 


0.8 

0.95 

1.1 


0.7 

0.85 

1.0 


0.65 

0.8 

0.95 


0.6 
0.7 
0.85 


0.6 
0.7 
0.8 


0.55 
0.65 
0.75 


200 
210 

i 220 
2 230 
g 240 
w 250 

^ 260 
fl 270 

280 


1.95 
2.25 
2.6 


1.8 

2.05 

2.35 


1.65 

1.9 

2.15 


1.5 
1.7 
1.95 


1.4 
1.6 
1.8 


1.25 
1.45 
1.65 


1.15 
1.35 
1.55 


1.1 

1.25 

1.45 


1.0 

1.15 

1.3 


0.95 

1.1 

1.25 


0.9 

1.05 

1.2 


2.95 

3.3 

3.75 


2.7 

3.05 

3.45 


2.45 

2.8 

3.15 


2.25 
2.55 
2.9 


2.05 
2.35 
2.65 


1.9 

2.15 

2.45 


1.75 

2.0 

2.25 


1.65 
1.85 
2.1 


1.5 

1.75 

1.95 


1.45 
1.65 
1.85 


1.35 
1.5 
1.7 


4.25 
4.75 
5.3 


3.85 

4.3 

4.8 


3.5 

3.95 

4.4 


3.25 

3.6 

4.0 


3.0 

3.35 

3.7 


2.75 

3.1 

3.45 


2.55 
2.85 
3.15 


2.35 
2.65 
2.95 


2.2 

2.45 

2.75 


2.05 

2.3 

2.55 


1.9 

2.15 

2.4 


f 290 
« 800 
1 310 


5.85 
6.45 
7.1 


5.3 

5.85 
6.45 


4.85 
5.35 
5.9 


4.45 

4.9 

5.4 


4.1 
4.5 
5.0 


3.8 
4.2 
4.6 


3.5 
3.9 

4.3 


3.25 
3.6 
3.95 


3.05 
3.35 
3.7 


2.85 
3.15 
3.45 


2.65 
2.95 
3.25 


o 320 
330 
340 


7.8 
8.5 
9.3 


7.1 

7.75 

8.45 


6.5 
7.1 
7.75 


5.95 

6.5 

7.1 


5.5 
6.0 
6.55 


5.05 
5.55 
6.05 


4.7 

5.15 

5.6 


4.S5 
4.75 
5.2 


4.05 
4.45 

4.85 


3.8 

4.15 

4.55 


3.55 

3.9 

4.25 


350 
360 
370 


10.1 

10.95 

11.85 


9.2 
10.00 
10.8 


8.4 
9.15 
9.9 


7.75 
8.4 
9.1 


7.7o 
8.4 


6.6 

7.15 
7.75 


6.1 

6.65 

7.2 


5.65 
6.15 
6.65 


5.3 

5.75 

6.2 


4.95 
5.35 
5.8 


4.6 
5.0 
5.45 


380 

390 
400 


12.8 
13.8 
14.85 


11.7 
12.6 
13.55 


10.7 
11.5 
12.4 


9.85 
10.6 
11.4 


9.05 
9.75 
10.5 


8.4 
9.05 
9.75 


7.8 
8.4 
9.05 


7.2 
7.8 
8.4 


6.7 
7.25 

7.85 


6.3 
6.8 
7.3 


5.9 

6.35 

6.85 


410 
420 
430 


15.9 

17.05 

18.2 


14.5 

15.55 

16.65 


13.3 

14.25 

15.25 


12.25 
13.15 
14.05 


11.3 
12.1 
12.95 


10.45 

11.2 

12.0 


9.7 
10.4 
11.15 


9.05 

9.7 

10.4 


8.4 
9.05 
9.7 


7.85 
8.45 
9.05 


7.4 

7.95 

8.5 


440 
450 
460 


19.45 

20.7 

22.05 


17.75 
18.95 
20.15 


16.3 

17.35 

18.5 


15.0 
16.0 
17.05 


13.85 
14.75 
15.75 


12.85 

13.7 

14.6 


11.9 
12.7 
13.5 


11.1 

11.85 

12.65 


10.35 
11.05 
11.8 


9.7 
10.35 
11.0 


9.05 
9.7 
10.35 


470 
480 
490 
500 


23.45 
24.85 
26.35 
27.9 


21.4 
22.7 
24.05 
25.45 


19.65 
20.9 
22.2 
23.55 


18.10 
19.25 
20.4 
21.55 


16.8 
17.85 
18.9 
19.95 


15.5 
16.45 
17.5 
18.55 


14.4 
15.3 
16.25 
17.2 


13.45 
14.25 
15.15 
16.05 


12.55 
13.3 
14.15 
15.0 


11.7 
12.4 
13.15 
13.95 


11.0 
11.7 
12.4 
13.2 



APPENDIX A. 



823 



TABLE II. — Continved. 
Eeflecting Galvanometer Scale Errors. 

(These corrections are to be subtracted from the observed deflections.) 



1550 


1600 


1650 


1700 


1750 


1800 


1850 


1900 


1950 


2000 


j Distance 


0. 
0. 
0.05 


0. 
0. 
0.05 


0. 
0. 
0.05 


0. 
0. 
0.05 


0. 
0. 
0.05 


0. 
0. 
0.05 


0. 
0. 
0.05 


0. 
0. 
0.05 


0. 
0. 
0. 


0. 
0. 
0. 


60 
60 
70 


0.05 
0.1 
0.1 


0.05 
0.05 
0.1 


0.05 
0.05 
0.1 


0.05 
0.05 
0.1 


0.05 
0.05 
0.1 


0.05 
0.l'5 
0.1 


0.05 
0.05 
0.05 


0.05 
0.05 
0.05 


0.05 
0.05 
0.05 


0.05 
0.05 
0.05 


80 

90 

100 


0.15 

0.2 

0.25 


0.15 
0.15 
0.2 


0.1 

0.15 

0.2 


0.1 

0.15 

0.2 


0.1 

0.15 

0.2 


0.1 

0.15 

0.15 


0.1 

0.15 

0.15 


0.1 
0.1 
0.15 


0.1 
0.1 
0.15 


0.1 
0.1 
0.15 


110 
120 
130 


0.3 

0.35 

0.45 


0.25 
0.35 
0.4 


0.25 

0.3 

0.4 


0.25 

0.3 

0.35 


0.25 

0.3 

0.35 


0.2 

0.25 

0.3 


0.2 

0.25 

0.3 


0.2 

0.25 

0.3 


0.2 
0.2 
0.25 


0.15 
0.2 
0.25 


140 
150 
160 


0.5 

0.6 

0.7 


0.5 

0.55 

0.65 


0.45 
0..55 
0.65 


0.45 

0.5 

0.6 


0.4 
0.5 
0.55 


0.4 
0.45 
0.55 


0.35 

0.4 

0.5 


0.35 

0.4 

0.5 


0.3 

0.4 
0.45 


0.3 

0.35 

0.45 


170 
180 
190 


0.8 

0.95 

1.1 


0.8 
0.9 
1.05 


0.75 
0.85 
1.0 


0.7 

0.8 
0.9 


0.65 
0.75 
0.85 


0.6 
0.7 
0.8 


0.6 

0.7 
0.8 


0..55 
0.65 
0.75 


0..55 
0.6 
0.7 


0.5 
0.6 
0.65 


200 
210 o 

220 g 


1.25 
1.45 
1.6 


1.2 

1.35 

1.5 


1.1 

1.25 

1.4 


1.05 

1.2 

1.35 


1.0 

1.15 

1.3 


0.95 
1.05 
1.2 


0.9 
1.0 
1.15 


0.85 
0.95 
1.05 


0.8 
0.9 
1.0 


0.75 
0.85 
1.0 


230 « 
240 < 
250 g 


1.8 
2.0 
2.25 


1.7 
1.9 
2.1 


1.6 
1.8 
2.0 


1.5 
1.7 

1.9 


1.45 

1.6 

1.75 


1.35 

1.5 

1.65 


1.3 

1.45 

1.6 


1.2 

1.35 

1.5 


1.15 
1.3 
1.45 


1.10 
1.25 
1.35 


260 W 
270 " 
280 1 

290 % 
300 3 
310 § 


2.5 

2.75 

3.05 


2.35 

2.6 

2.85 


2.2 
2.4 
2.65 


2.1 
2.3 
2.5 


1.95 
2.15 
2.35 


1.85 
2.05 
2.25 


1.75 
1.95 
2.15 


1.65 
1.85 
2.05 


1.6 

1.75 

1.95 


1.5 
1.7 
1.85 


3.35 
3.65 
4.0 


3.15 
3.45 

3.75 


2.95 
3.25 
3.5 


2.75 
3.05 
3.35 


2.6 
2.85 
3.1 


2.45 

2.7 

2.95 


2.35 
2.55 

2.8 


2.25 
2.45 
2.65 


2.15 
2.35 
2.55 


2.05 

2.2 

2.4 


320 
330 
340 


4.35 

4.7 

5.1 


4.1 

4.45 
4.8 


3.85 

4.2 

4.55 


3.65 
3.95 
4.3 


3.4 
3.7 
4.0 


3.25 
3.55 
3.85 


3.05 
3.35 
3.65 


2.9 

3.15 

3.40 


2.75 

3.0 

3.25 


2.6 
2.85 
3.1 


350 
360 
370 


5.5 

5.95 

6.45 


5.2 
5.6 
6.05 


4.9 
5.3 
5.7 


4.65 

5.0 

5.35 


4.35 

4.7 

5.05 


4.15 
4.45 
4.80 


3.95 
4.25 
4.55 


3.7 
4.0 
4.3 


3.5 
3.8 
4.1 


3.35 
3.6 
3.9 


380 
390 
400 


6.95 
7.45 
7.95 


6.5 
7.0 
7.5 


6.15 
6.6 
7.05 


5.75 

6.2 

6.65 


5.45 

5.85 
6.25 


5.2 

5.55 

5.95 


4.9 
5.3 
5.65 


4.65 

5.0 

5.35 


4.4 

4.75 

5.1 


4.2 
4.5 

4.85 


410 
420 
430 


8.5 
9.1 
9.7 


8.0 

8.55 

9.1 


7.55 

8.05 
8.6 


7.1 

7.6 
8.1 


6.7 

7.15 
7.65 


6..35 

6.8 

7.25 


6.05 
6.45 
6.85 


5.75 
6.15 
6.55 


5.45 

5.8 

6.2 


5.2 

5.55 

5.9 


440 
450 
460 


10.3 
10.95 
11.65 
12.35 


9.7 
10.35 
11.0 
11.65 


9.15 
9.75 
10.35 
10.95 


8.65 

9.2 

9.75 

10.35 


8.15 

8.7 

9.25 

9.8 


7.75 

8.25 
8.75 
9.3 


7.3 
7.75 
8.25 
8.75 


6.95 

7.4 
7 85 
8.35 


6.6 
7.0 
7.45 
7.9 


6.3 
6.7 
7.1 
7.55 


470 
480 
490 
500 



•324 



ELECTRICAL MEASUREMENTS. 



TABLE III. 

Reduction of the Period to an Infinitely Small Arc. 

If the observed time of oscillation be I^, with an arc 
of oscillation of a degrees, cTmust be subtracted from 
the observed value to reduce to an infinitely small arc 
of oscillation. 



a 


c 






c 




a 


c 




a 


c 




0° 


0.00000 




10° 


0.00048 




20« 


0.00190 




30" 


0.00428 




1 


000 





11 


058 


10 


21 


210 


20 


31 


457 


29 


2 


• 002 


2 


12 


069 


11 


22 


230 


20 


32 


487 


30 


3 


004 


2 


13 


080 


11 


23 


251 


21 


33 


518 


31 


4 


008 


4 


14 


093 


13 


24 


274 


23 


34 


550 


32 


5 


012 


4 


15 


107 


14 


25 


297 


23 


35 


583 


33 


6 


017 


5 


16 


122 


15 


26 


322 


25 


36 


616 


33 


7 


023 


6 


17 


138 


16 


97 


347 


25 


37 


651 


35 


8 


030 


7 


18 


154 


16 


28 


373 


26 


38 


686 


35 


9 


039 


9 


19 


172 


18 


29 


400 


27 


39 


723 


37 


10 


0.00048 


9 


20 


0.00190 


18 


30 


0.00428 


28 


40 


0.00761 


38 



TABLE IV. 
E.M.F. of Standard Cells at Different Temperatures. 





Clark 


Cell. 




Carhakt-Clark Cell. 


Temp., C. 


E.M.F. 


Temp., C. 


E.M.F. 


Temp.,C. 


E.M.F. 


Temp., C. 


E.M.F. 


10° 


1.4396 


20O 


i 
1.4279 I 


i lO" 


1.4428 


20° 


1.4372 


11 


1.4385 


21 


1.4267 


1 11 


1.4422 


21 


1.4367 


12 


1.4374 


22 


1.4254 


12 


1.4417 


22 


1.4361 


13 


1.4363 


23 


1.4241 


13 


1.4411 


23 


1.4356 


14 


1.4352 


24 


1.4227 


14 


1.4406 


24 


1.4350 


15 


1.4340 


25 


1.4214 


15 


1.4400 


25 


1.4345 


16 


1.4328 


26 


1.4200 


16 


1.4394 


26 


1.4340 


17 


1.4316 


27 


1.41S6 


17 


1.4389 


27 


1.4334 


18 


1.4304 


28 


1.4172 


18 


1.4383 


28 


1.4329 


19 


1.4292 


29 


1.4158 


19 


1.4378 


29 


1.4.323 






30 


1.4143 






30 


1.4318 



APPEXDIX A. 


^^5 


TABLE V. 




Dimensional Formulas. 




I. yieehanical Units. 




Area 


L' 


Volume . 


. 


U 


Velocity . 




LT-' 


Acceleration 




LT-'- 


Force 




LMT-'- 


Moment of rotation 




UMT-' 


Moment of inertia 




UM 


Work, energy . 




UMT-' 


IL Electric Units. 




Quantity of electricity 


M\L\ 


Electric potential, electromotive force 


. mmT-' 


Capacity 


L-'T' 


Current strength .... 


M^L^T-' 


Resistance 


LT-' 


Inductance 


L 


III. Magnetic Units. 




Strength of pole .... 


. M^UT-' 


Magnetic moment .... 


. M^nT-' 


Intensity of magnetization 


. M^L-'^T-' 


Magnetic force, intensity of held 


. M^L-\T-' 



The above electric and magnetic units are in the electro- 
magnetic system. 



326 



ELECTRICAL ME A S UBEMENTS. 



TABLE VI. 
Doubled Square Roots for Kelvin Balances. 








100 


200 


300 


400 


600 


600 


700 


800 


900 







0-000 


20-00 


28-28 


34-64 40-00 


44-72 


48-99 52-92 


56-57 


60-00 





1 


2-000 


20-10 


28-35 


34-70 40-05 j 


44-77 


49-03 I 52-95 


56-60 


60-03 


1 


2 


2-828 


20-20 


28-43 


34-76 40-10 ' 


44-81 


49-07 


52-99 


56-64 


60-07 


2 


3 


3-464 


20-30 


28-50 


34-81 40-15 


44-86 


49-11 


53-03 


56-67 : 60-10 


3 


4 


4-000 


20-40 


28-. 57 


34-87 40-20 


44-90 


49-15 


53-07 


56-71 1 60-13 


4 


5 


4-472 


20-49 


28-64 


34-93 


40-25 


44-94 


49-19 


53-10 


56-75 1 60-17 


5 


6 


4-899 


20-59 


28-71 


34-99 


40-30 


44-99 


49-23 


53-14 


56-78 60-20 


6 


7 


5-292 


20-69 


'28-77 


35-04 


40-35 


45-03 


49-27 


53-18 


56-82 i 60-23 


7 


8 


5-657 


20-78 


28-84 


35-10 


40-40 ! 


45-08 


49-32 


53-22 


56-85 


60 27 


8 


9 


6-000 


20-88 


28-91 


35-16 


40-45 


45-12 


49-36 


.53-25 


56-89 


60-30 


9 


10 


6-325 


20-98 


28-98 


35-21 


40-50 


45-17 


49-40 


53-29 


56-92 


60-33 


10 


11 


6-633 


21-07 


29-05 


35-27 


40-55 


45-21 


49-44 


53-33 


56-96 


60-37 


11 


12 


6-928 


21-17 


29-12 


35-33 


40-60 


45-25 


49-48 


53-37 


56-99 


60-40 


12 


13 


7-211 


21-26 


29-19 


35-38 


40-64 


45-30 


49-52 


53-40 


57-03 


60-43 


13 


14 


7-483 


21-35 


29-26 


35-44 


40-69 ! 


45-34 i 49-56 


53-44 


57-06 


60-46 


14 


15 


7-746 


21-45 


29-33 


35-50 


40-74 ; 
40-79 


45-39 49-60 


53-48 


57-10 


60-50 


15 


16 


8-000 


21-54 


29-39 


35-55 


45-43 49-64 


53-52 


57-13 


60-53 


16 


17 


8-246 


21-63 


29-46 


35-61 


40-84 


45-48 1 49-68 


53-55 


57-17 


60-56 


17 


18 


8-485 


21-73 


29-.53 


35-67 


40-89 


45-52 1 49-72 


53-59 


57-20 


60-60 


18 


19 


8-718 


21-82 


29-60 


35-72 


40-94 


45-56 


49-76 i 53-63 


57-24 


60-63 


19 


20 


8-944 


21-91 


29-66 


35-78 


40-99 


45-61 


49-80 


53-67 


57-27 


60-66 


2C 


21 


9-165 


22-00 


29-73 


35-83 


41-04 


45-65 


49-84 


.53-70 


57-31 


60-70 


21 


22 


9-381 


22-09 


29-80 


35-89 


41-09 


45-69 


49-88 


53-74 


57-34 


60-73 


22 


23 


9-592 


22-18 


29-87 


.35-94 


41-13 


45-74 


49-92 


53-78 


57-38 


60-76 


23 


24 


9-798 


22-27 


29-93 


36-00 


41-18 j 


45-78 


49-96 


53-81 


57-41 


60-79 


24 


25 


10-000 


22-36 


30-00 


36-06 


41-23 


45-83 


50-00 


53-85 


57-45 


60-83 


25 


26 


10198 


22-45 


30-07 


36-11 


41-28 ' 


45-87 


50-04 


53-89 


57-48 


60-86 


26 


27 


10-392 


22-54 


30-13 


36-17 


41-.33 , 


45-91 


50-08 


53-93 


57-52 


60-89 


27 


28 


10-583 


22-63 


30-20 


36-22 


41-38 


45-96 j 50-12 


53-96 


57-55 


60-93 


28 


29 


10.770 


22-72 


30-27 


36-28 


41-42 1 


46-00 ! 50-16 


54-00 


57-58 


60-96 


29 


30 


10-954 


22-80 


30-33 


36-33 


41-47 1 


46-04 1 50-20 

1 


54-04 


57-62 


60-99 


30 


31 


11-136 


•22-89 ! 30-40 


36-39 


41-52 


46-09 


50-24 


54-07 


57-65 


61-02 


31 


32 


11-314 


22-98 30-46 


36-44 


41-57 


46-13 


50-28 


54-11 


57-69 


61-06 


32 


33 


11-489 


23-07 1 30-53 


36-50 


41-62 


46-17 


50-32 


54-15 


57-72 


61-09 


33 


34 


11-662 


23-15 ! 30-59 


36-. 55 


41-67 


46-22 


50-36 


54-18 


57-76 


61-12 


34 


35 


11-832 


23-24 


30-66 


36-61 


41-71 


46-26 


50-40 


54-22 


57-79 


61-16 


35 


36 


12-000 


23-32 


30-72 


36-66 


41-76 


46^30 


50-44 


54-26 


57-83 


61-19 


36 


37 


12-166 


•23-41 


30-79 j 36-72 


41-81 


46-35 


50-48 


54-30 


57-86 


61-22 


37 


38 


12-329 


23-49 


30-85 36-77 ! 41-86 


46-39 


.50-52 


54-33 


57-90 


61-25 


38 t 


39 


12-490 


23-58 


30-92 1 36-82 


41-90 , 


46-43 


50-56 


54-37 


57-93 


61-29 


39 


40 


12-649 


23-66 


30-98 36-88 


41-95 1 


46-48 


50-60 


54-41 


57-97 


61-32 


40 


41 


12-806 


23.75 


31-05 36-93 


42-00 


46-52 


50-64 


54-44 


58-00 


61-35 


41 


42 


12-961 


23-83 


31-11 1 36-99 


42-05 


46-56 


50-6S 


54-48 


58-03 


61-38 


42 


43 


13-115 


•23-92 


31-18 : 37.04 


42-10 


46-60 


50-71 


.54-52 


58-07 


61-42 


43 


44 


13-266 


24-00 


31-24 i 37-09 


42-14 


46-65 


50-75 


54-55 58-10 


61-45 


44 


45 


13-416 


24-08 


31-30 1 37-15 


42-19 [ 


46-69 i 50-79 


54-59 58-14 


61-48 


45 


46 


13-565 


24-17 


31-37 37-20 


42-24 i 


46-73 50-83 


54-63 


58-17 


61-51 


46 


47 


13-711 


•24-25 


31-43 37-26 


42-28 ! 


46-78 50-87 


54-66 


58-21 


61-55 


47 


48 


13-856 


24-33 


31-50 37-.31 


42-33 1 


46-82 .50-91 


54-70 


58-24 


61-58 


48 


49 


14-000 


•24^41 


31-.56 ; 37-36 


42-38 


46-86 ; 50-95 


54-74 


58-28 


61-6 


49 


50 


14-142 


24-49 


31-62 


37-42 


42-43 


46-90 


50-99 


54-77 


68-31 


61-64 


50 



APPENDIX A. 



327 



TABLE VI. — Continued. 
Doubled Square Boots for Kelvin Balances. 








100 


200 


300 


400 


51 


14-283 


24-58 


31-69 


37-47 


42-47 


52 


14-422 


24-66 


31-75 


37-52 


42-52 1 


53 


14-560 


24-74 


31-81 


37-58 


42-57 


54 


14-697 


24-82 


31-87 


37-63 


42-61 


55 


14-832 


24-90 


31-94 


37-68 


42-66 


56 


14-967 


24-98 


32-00 


37-74 


42-71 


57 


15-100 


25-06 


32-06 


37-79 


42-76 


58 


15-232 


25-14 


32-12 


37-84 


42-80 


59 


15-362 


25-22 


32-19 


37-89 


42-85 


60 


15-492 


25-30 


32-25 


37-95 


42-90 


61 


15.620 


25-38 


32-31 


38-00 


42-94 


62 


15-748 


25-46 


32-37 


38-05 


42-99 


63 


15-875 


25-53 


32-43 


38-11 


43-03 


64 


16-000 


•25-61 


32-50 


38-16 


43-08 


65 


16-125 


•25-69 


32-56 


38-21 43-13 


66 


16-248 


•25-77 


32-62 


38-26 


43-17 


67 


16-371 


25-85 


32-68 


38-31 


43-22 


68 


16-492 


25-92 


32-74 


38-37 


4.3-27 


69 


16-613 


26-00 


32-80 


38-42 


43-31 


70 


16-733 


26-08 


32-86 


38-47 


43-36 


71 


16-852 


26-15 


32-92 


38-52 


43-41 


72 


16-971 


26-23 


32-98 


38-57 


43-45 


73 


17-088 


26-31 


33-05 


38-63 


43-50 


74 


17-205 


^6-38 


33-11 


38-68 


43-54 


75 


17-321 


26-46 


33-17 


38-73 


43-59 


76 


17-436 


26-53 


33-23 


38-78 


43-63 


77 


17-550 


26-61 


33-29 


38-83 


43-68 


78 


17-664 


26-68 


33-35 


38-88 


43-73 


79 


17-776 


26-76 


33 41 


38-94 


43-77 


80 


17-889 


26-83 


33-47 


38-99 


43-82 


81 


18.000 


26-91 


33-53 


39-04 


43-86 


82 


18-111 


26-98 


33-59 


39-09 


43-91 


83 


18-221 


27-06 


33-65 


39-14 


43-95 1 


84 


18-3.30 


27-13 


33^70 


39-19 


44-00 i 


85 


18-439 


27-20 


33-76 


39-24 


44-05 


86 


18-547 


27-28 


33-82 


39-29 


44^09 


87 


18-655 


•27-35 


.3.3-88 


39-34 


44-14 


88 


18-762 


27-42 


33-94 


39-40 


44-18 


89 


18-868 


27-50 


34-00 


39-45 


44-23 


90 


18-974 


27^57 


34-06 


39-50 


44-27 


91 


19-079 


27-64 


34-12 


39-55 


44-32 


92 


19-183 


27-71 


34-18 


39-60 


44-36 


93 


19-287 


27-78 


34-23 


39-65 


44-41 


94 


19-391 


27-86 


34-29 


36-70 


44-45 


95 


19-494 


27-93 


34-35 


39-75 


44-50 


96 


19-596 


28-00 


34-41 


39-80 


44-54 i 


97 


19-698 


28-07 


34-47 


39-85 


44-59 


98 


19-799 


28-14 


34-53 


.39-90 


44-63 


99 


19-900 


•28-21 


34-58 


39-95 


44-68 


100 


20-000 


•2S-28 


34-64 


40-00 


44-72 



500 


600 


700 


800 


900 




46-95 


51-03 


54-81 


58-34 


61-68 


51 


46-99 


51-07 


54-85 


58-38 


61-71 


52 


47-03 


51-11 


54-88 


58-41 


61-74 


53 


I 47-07 


51-15 


54-92 


58-45 


61-77 


54 


4712 


.51-19 


54-95 


58-48 


61-81 


55 


47-16 


51-22 


54-99 


58-51 


61-84 


56 


47-20 


51-26 


55-03 


58-55 


61-87 


57 


47-24 


51-30 


55.06 


58-58 


61-90 


58 


47^29 


51-34 


55-10 


58-62 


61-94 


59 


47-33 


51-38 


55-14 


58-65 


61-97 


60 


47-37 


.51-42 


55-17 


58-69 


62-00 


61 


47-41 


51-46 


55-21 


58-72 


62-03 


62 


47-46 


51-50 


55-24 


58-75 


62-06 


63 


47-50 


51-54 


55-28 


58-79 


6-2^10 


64 


47-54 


51-58 


55-32 


58-82 


62-13 


65 


47-58 


51-61 


55-35 


58-86 


62-16 


66 


47-62 


51-65 


55-39 


58-89 


62-19 


67 


47-67 


51-69 


55-43 


58-92 


62-23 


68 


47-71 


51-73 


55-46 


58-96 


62^26 


69 


47-75 


51-77 


55-50 


58-99 


62-29 


70 


47-79 


51-81 


55-53 


.59-03 


62-32 


71 


47-83 


51-85 


55-57 


59-06 


62-35 


72 


47-87 


51-88 


.55-61 


59-09 


62-39 


73 


47-92 


51-92 


.55-64 


59-13 


62-42 


74 


47^96 


51-96 


55-68 


59-16 


62-45 


75 


48-00 


52-00 


55-71 


59-19 


62-48 


76 


48-04 


52-04 


55-75 


59-23 


62-51 


77 


48-08 


52-08 


.55-79 


59-26 


62-55 


78 


48-12 


52-12 


55^82 


59-30 


62-58 


79 


48-17 


52-15 


55-86 


59-33 


62-61 


80 


48-21 


52-19 


55-89 


59-36 


62-64 


81 


48-25 


52-23 


55-93 


59-40 


62-67 


82 


48-29 


52-27 


55-96 


59-43 


62-71 


83 


48-33 


52-31 


56-00 


59-46 


62-74 


84 


48-37 


52-35 


56-04 


59-50 


62-77 


85 


48-41 


52-38 


56-07 


.59-53 


62-80 


86 


48-46 


52-42 


56-11 


59.57 


62-83 


87 


48-50 


52-46 


56-14 


59-60 


62-86 


88 


48-54 


52-50 


56-18 


59-63 


62-90 


89 


48-58 


52-54 


56-21 


59-67 


62-93 


90 


48-62 


52-57 


56-25 


59-70 


62-96 


91 


48-66 


52-61 


56-28 


59-73 


62-99 


92 


48-70 


5-2-65 


56-32 


59-77 


63-02 


93 


48-74 


52-69 


56-36 


59-80 


63-06 


94 


48-79 


5-2-73 


56-39 


59-83 


63-09 


95 


48-83 


52-76 


56-43 


59-87 


63-12 


96 


48-87 


5-2-80 


56-46 


59-90 


63-15 


97 


48-91 


52-84 


56-50 


59-93 


63-18 


98 


48-95 


52-88 


56-53 


59-97 


63-21 


99 


48-99 


52-92 


56-57 


60-00 


63-25 


100 



328 ELECTRICAL MEASUREMENTS. 



APPENDIX B. 



SPECIFICATIONS FOR THE PRACTICAL APPLICA- 
TION OF THE DEFINITIONS OF THE AMPERE 
AND VOLT.i 

SPECIFICATION A. -The Ampere. 

In employing the silver voltameter to measure currents 
of about one ampere, the following arrangements shall 
be adopted: 

The kathode on Avhich the silver is to be deposited 
shall take the form of a platinum bowl not less than 10 
cms. in diameter, and from 4 to 5 cms. in depth. 

The anode shall be a disc or plate of pure silver some 
30 sq. cms. in area, and 2 or 3 cms. in thickness. 

This shall be supported horizontally^ in the liquid near 
the top of the solution by a silver rod riveted through 
its centre. To prevent the disintegrated silver which is 
formed on the anode from falling upon the kathode, the 
anode shall be wrapped around with pure filter paper, 
secured at the back by suitable folding. 

The liquid shall consist of a neutral solution of pure 
silver nitrate, containing about fifteen parts by weight 
of the nitrate to 85 parts of water. 

The resistance of the voltameter changes somewhat as 
the current passes. To prevent these changes having 
too great an effect on the current some resistance, 
besides that of the voltameter, should be inserted in the 

1 Legalized by act of Congress, approved July 12, 1894. 



APPENDIX B. 329 

circuit. The total metallic resistance of the circuit 
should not be less than 10 ohms. 

Method of making a Measurement. — The platinum 
bowl is to be washed consecutively with nitric acid, 
distilled water, and absolute alcohol ; it is then to be 
dried at 160° C, and left to cool in a desiccator. When 
thoroughly cool it is to be weighed carefully. 

It is to be nearly filled with the solution and con- 
nected to the rest of the circuit by being placed on 
a clean copper support to which a binding screw is at- 
tached. 

The anode is then to be immersed in the solution so 
as to be well covered by it, and supported in that position ; 
the connections to the rest of the circuit are then to be 
made. 

Contact is to be made at the key, noting the time. 
The current is to be allowed to pass for not less than 
half an hour, and the time of breaking contact observed. 

The solution is now to be removed from the bowl, and 
the deposit washed with distilled water, and left to soak 
for at least six hours. It is then to be rinsed successively 
with distilled water and absolute alcohol, and dried in a 
hot-air bath at a temperature of about 160° C. After 
cooling in a desiccator it is to be weighed again. The 
gain in mass gives the silver deposited. 

To find the time average of the current in amperes, 
this mass, expressed in grammes, must be divided by the 
number of seconds .during which the current has passed 
and by 0.001118. 

In determining the constant of an instrument by this 
method the current should be kept as nearly uniform as 
possible, and the readings of the instrument observed at 
frequent intervals of time. These observations give a 



330 ELECTRICAL MEASUREMENTS. 

curve from which the reading corresponding to the mean 
current (time-average of the current) can be found. 
The current, as calculated from the voltameter results, 
corresponds to this reading. 

The current used in this experiment must be obtained 
from a battery and not from a dynamo, especially when 
the instrument to be calibrated is an electrodyna- 
mometer. 

SPECIFICATION B.- The Volt. 

Definition and Properties of the Cell. — The cell has 
for its positive electrode, mercury, and for its negative 
electrode, amalgamated zinc ; the electrolyte consists of 
a saturated solution of zinc sulphate and mercurous 
sulphate. The electromotive force is 1.-434 volts at 
15° C, and, between 10° C. and 25° C, by the increase 
of 1° C. in temperature, the electromotive force decreases 
by .00115 of a volt. 

1. Preparation of the Mercury. — To secure purity 
it should be fu'st treated with acid in the usual manner 
and subsequently distilled in vacuo. 

2. Preparation of the Zinc Araalg-am. — The zinc 
designated in commerce as '' commercially pure " can 
be used without further preparation. For the prepara- 
tion of the amalgam one part by weight of zinc is to be 
added to nine (9) parts by weight of mercury, and both 
are to be heated in a porcelain dish at 100° C. with 
moderate stirring until the zinc has ^een full}- dissolved 
in the mercur}-. 

3. Preparation of the Mercurous Sulphate. — Take 
mercurous sulphate, purchased as pure, mix with it a 
small quantity of pure mercur}-, and wash the whole 
thoroughly Avith cold distilled water by agitation in a 



APPENDIX B. 331 

bottle ; drain off the water and repeat the process at 
least twice. After the last washing drain off as much 
of the water as possible. (For further details of puri- 
fication, see Note A.) 

4. Preparation of the Zinc Sulphate Solution. — Pre- 
pare a neutral saturated solution of pure re-crystallized 
zinc sulphate, free from iron, by mixing distilled water 
with nearly twice its weight of crystals of pure zinc 
sulphate and adding zinc oxide in the proportion of 
about 2 per cent by weight of the zinc sulphate crystals 
to neutralize any free acid. The crystals should be dis- 
solved with the aid of gentle heat, but the temperature 
to which the solution is raised must not exceed 30° C. 
Mercurous sulphate, treated as described in 3, shall be 
added in the proportion of about 12 per cent by weight 
of the zinc sulphate crystals to neutralize the free zinc 
oxide remaining, and then the solution filtered, while 
still warm, into a stock bottle. Crystals should form as 
it cools. 

5. Preparation of the Mercurous Sulphate and Zinc 

Sulphate Paste For making the paste, two or three 

parts by weight of mercurous sulphate are to be added 
to one by weight of mercury. If the sulphate be dry, 
it is to be mixed with a paste consisting of zinc sulphate 
crystals and a concentrated zinc sulphate solution, so 
that the Avhole constitutes a stiff mass, which is 
permeated throughout by zinc sulphate crystals and 
globules of mercury. If the sulphate, however, be 
moist, only zinc sulphate crystals are to be added; care 
must, however, be taken tliat tliese occur in excess and 
are not dissolved after continued standing. The mer- 
cury must, in this case also, permeate the paste in little 
globules. It is advantageous to crush the zinc sulphate 



332 ELECTRICAL MEASUREMENTS. 

crystals before using, since the paste can then be better 
manipulated. 

To set up the Cell, — The containing glass vessel, 
represented in the accompanying figure,^ shall consist 
of two limbs closed at bottom and joined above to a 
common neck fitted with a ground-glass stopper. The 
diameter of the limbs should be at least 2 cms. and their 
length at least 3 cms. The neck should be not less than 
1.5 cm^. in diameter. At the bottom of each limb a 
platinum wire of about 0.4 mm. diameter is sealed 
through the glass. 

To set up the cell, place in one limb mercury and in 
the other hot liquid amalgam, containing 90 parts mer- 
cury and 10 parts zinc. The platinum wires at the 
bottom must be completely covered by the mercury and 
the amalgam respectively. On the mercury, place a 
layer one cm. thick of the zinc and mercurous sulphate 
paste described in 5. Both this paste and the zinc 
amalgam must then be covered with a layer of the 
neutral zinc sulphate crystals one cm. thick. The whole 
vessel must then be filled with the saturated zinc sul- 
phate solution, and the stopper inserted so that it shall 
just touch it, leaving, however, a small bubble to guard 
against breakage when the temperature rises. 

Before finally inserting the glass stopper, it is to be 
brushed round its upper edge with a strong alcoholic 
solution of shellac and pressed fumly in place. (For 
details of filling the cell, see Note B.) 



See Fig. 85, page 178. 



APPENDIX B. 333 



NOTES TO THE SPECIFICATIONS. 

(A.) The Mercurous Sulphate. — The treatment of 
the mercurous sulphate has for its object the removal 
of any mercuric sulphate which is often present as an 
impurity. 

Mercuric sulphate decomposes in the presence of 
water into an acid and a basic sulphate. The latter is a 
yellow substance — turpeth mineral — practically in- 
soluble in water ; its presence, at any rate in moderate 
quantities, has no effect on the cell. If, however, it be 
formed, the acid sulphate is also formed. This is soluble 
in water and the acid produced affects the electromotive 
force. The object of the washings is to dissolve and 
remove this acid sulphate, and for this purpose the three 
washings described in the specification will suffice in 
nearly all cases. If, however, much of the turpeth 
mineral be formed, it shows that there is a great deal of 
the acid sulphate present, and it will then be wiser to 
obtain a fresh sample of mercurous sulphate, rather than 
to try by repeated washings to get rid of all the acid. 

The free mercury helps in the process of removing the 
acid^ for the acid mercuric sulphate attacks it, forming 
mercurous sulphate. 

Pure mercurous sulphate, when quite free from acid, 
shows on repeated washing a faint yellow tinge, which 
is due to the formation of a basic mercurous salt distinct 
from the turpeth mineral, or basic mercuric sulphate. 
The appearance of this primrose yellow tint ma}^ be 
taken as an indication that all the acid has been re- 
moved; the washing may with advantage be continued 
until this tint appears. 



334 ELECTBICAL MEASUREMENTS. 

(B.) Pilling' the Cell. — After thoroughly cleaning 
and drying the glass vessel, place it in a hot-water bath. 
Then pass through the neck of the vessel a thin glass 
tube reaching to the bottom to serve for the introduction 
of the amalgam. This tube should be as large as the 
glass vessel will admit. It serves to protect the upper 
part of the cell from being soiled with the amalgam. 
To fill in the amalgam, a clean dropping tube about 10 
cms. long, drawn out to a fine point, should be used. Its 
lower end is brought under the surface of the amalgam 
heated in a porcelain dish, and some of the amalgam is 
drawn into the tube by means of the rubber bulb. The 
point is then quickly cleaned of dross with filter paper 
and is passed through the wider tube to the bottom 
and emptied by pressing the bulb. The point of the 
tube must be so fine that the amalgam will come out 
only on squeezing the bulb. This process is repeated 
until the limb contains the desired quantity of the amal- 
gam. The vessel is then removed from the water- 
bath. After cooling, the amalgam must adhere to 
the glass and must show a clean surface with a metallic 
lustre. 

For insertion of the mercury, a dropping tube with a 
long stem will be found convenient. The paste may be 
poured in through a wide tube reaching nearly down 
to the mercury and having a funnel-shaped top. If the 
paste does not move down freely it may be pushed down 
with a small glass rod. The paste and the amalgam are 
then both covered with the zinc sulphate crystals before 
the concentrated zinc sulphate solution is poured in. 
This should be added through a small funnel, so as to 
leave the neck of the vessel clean and dry. 

For convenience and security in handling, the cell 



APPENDIX B. 335 

may be mounted in a suitable case so as to be at all 
times open to inspection. 

In using the cell, sudden variations of tempera- 
ture should, as far as possible, be avoided, since the 
changes in electromotive force lag behind those of tem- 
perature. 



INDEX 



Numbers refer to pages. 



Absolute capacity of a con- 
denser, 227, 229, 230. 

Absorption, correction for, 220. 

Acceleration, 7. 

Activity, 9. 

Air-Leyden, 214. 

Ampere, 16; international, 17, 
328. 

Anderson's mocliti cation of 
Maxwell's method, 249. 

Angle of lag, 236, 238. 

Archives, Icilogramme des, 6 ; 
metre des, 5. 

Arrangement for strong or 
weak currents, 168. 

Astatic galvanometer, 145. 

Auxiliary apparatus for measur- 
ing internal resistance, 106. 

Ayrton on d' Arsonval galvanom- 
eter, 136; insulation resist- 
ance, 83; and Perry's method 
for electrolytic resistance, 115. 

B. A. units and international 
units, 18. 

Balances, Kelvin (Thomson), 
141, 193. 

Ballistic galvanometer, 207 ; 
galvanometer, constant of, 88, 
309, 310, 318 ; method of mag- 
netic measurements, 307, 308, 



314, 316 ; compared with trac- 
tional method, 305, 316. 

Bar and yoke, Hopkinson's, 
314. 

Barcelona, metre des archives, 5. 

Bars, magnetic test of, 299, 303, 
306, 307, 308, 314. 

Barus, Carl, calibration of bridge 
wire, 78. 

Battery, internal resistance of 
{see Eesistance) . 

Bidwell, Shelf ord, divided ring- 
method, 306. 

Borda, hilogramme des archives, 
6 ; metre des archives, 5. 

Bosanquet, divided rod method, 
306. 

Box resistance, 25; Post-Office 
resistance, 48 ; shunt, 34. 

Bridge, conductivity, 82; meth- 
ods of comparing capacities, 
218, 219; comparing capacity 
and self-induction, 245, 253 ; 
comparing mutual and self- 



induction. 



comparm< 



self-inductions, 255, 258 ; meas- 
urement of absolute capacity, 
230; Post-Office, 48; slide 
wire, 51, 54, '^Q, 58, 64; Wheat- 
stone's, 45 ; wire, calibration 
of, 73, 78. 



338 



INDEX. 



British Association, C.G.S. sys- 
tem, 6; units, 16, 18. 
Broch, density of water, 6. 

Calibration of bridge wire, 73, 
78; of electrostatic voltmeter, 
202; of galvanometer, 37, 88, 
150, 151, 154, 309, 310, 318; of 
voltmeter by standard cells, 
205. 

Calomel, one-volt cell, 183. 

Capacity, 15, 207; absolute, of a 
condenser, 227, 229, 230; com- 
parison of, by bridge methods, 
218, 219; by divided charge, 
216; by Gott's method, 219; 
by Thomson's method of mixt- 
ures, 222 ; with self-induction, 
245, 249, 251, 253; measure- 
ment of by alternating cur- 
rents, 241 ; of an electrostatic 
voltmeter, 240; solution for 
current with both self-induc- 
tion and capacity, 237 ; static, 
of coils, 115, 244. 

Carhart-Clark standard cell, 181. 

Chaperon, static capacity of 
coils, 115. 

Chicago Congress of 1893, 16. 

Clark standard cell 18, 176, 324, 
330. 

Coefficient of mutual induction 
{see Mutual induction) ; of 
self-induction {see Self-induc- 
tion) ; resistance temperature, 
23, 80; E.M.F. temperature, 
180, 182, 324, 330. 

Coercive force, 280. 

Commutator, Pohl's, 28; double, 
109. 

Condenser, comparison of capac- 



ity {see Capacity) ; discharge 
through high resistance, 223; 
method of comparing 
E.M.F.'s, 188; of measuring 
internal resistance, 100 ; stand- 
ard, 213. 

Conductivity, 22 ; bridge, 82. 

Constant of current meter by 
electrolysis, 164, 329 ; of a gal- 
vanometer, 37, 88, 164, 309, 310, 
318. 

Control magnet, 32, 148. 

Copper, resistance temperature 
coefficient of, 23 ; voltameter, 
161. 

Correction, for absorption, 220; 
for bridge wire, 74, 75, 77, 80; 
for damping, 211, 310; of de- 
flections, 37, 321, 322; of 
E.M.F. of cells, 180, 182, 324, 
330; of periods to infinitely 
small arc, 324 ; of resistance 
for temperature, 23 ; for ends 
of a bar, 281 ; for induced mag- 
netization, 295. 

Cosine galvanometer, 126. 

Coulomb, 16; international, 18. 

Creeping up of magnetization, 
303, 312. 

Current, arrangement for strong 

or weak, 168 ; measurement of, 

118; by cosine galvanometer, 

126; by electrolysis, 156, 164, 

328; by Kelvin balances, 141, 

! 193; by electrodynamometer^ 

127 ; by standard cell, 169, 172 ; 

j plotting of, 121 ; strength of, 

I 12 ; variation of internal resist- 

' ance with, 104; with both 

self-induction and capacity, 

I 237. 



INDEX. 



339 



Cyclical magnetization curve, 
299, 311, 315. 

Damping, correction for, 211, 
310. 

Daniel, electrolytic resistance, 
113. 

D'Arsonval galvanometer, 57, 
135; best form of coil of, 
139. 

Deflections, scale, in terms of 
angle, tangent, etc., 37, 321, 
322. 

Delambre, metre des archives, 5. 

Demagnetization of rings and 
bars, 297, 302. 

Derived units, fundamental 
and, 1. 

Determination of p, Q&. 

Dewar, electrical resistance, 14. 

Difference of potential, 14. 

Differential galvanometer, resist- 
ance by, 40, 44. 

Dimensional formulas, 1, 325; 
use of, 3. 

Dip, magnetic, 282, 284 ; needle, 
282. 

Direct deflection, insulation re- 
sistance by, 86. 

Discharge of a condenser 
through high resistance, 223 ; 
residual, 225. 

Divided, charge, comparison of 
capacities by, 216 ; ring meth- 
od of magnetic measurements, 
305 ; rod method of magnetic 
measurements, 303, 306, 307. 

Double, commutator, 109 ; key, 
48. 

Doubled square roots, 144; table 
of, 326. 



Du Bois, optical magnetic meth- 
od, 299. 
Dunkirk, metre des archives, 5. 
Dyne, 8. 

Earth-inductor, 284, 309. 

Earth's magnetic field, 119; ef- 
fect on electrodynamometer, 
130. 

Electrical units, magnetic and, 
9, 325 ; two systems of, 11. 

Electrodes, 156. 

Electrodynamometers, Siemens, 
127; afiected by earth's field, 
130. 

Electrolysis, measurement of 
current by, 156 ; determination 
of constant by, 164, 329. 

Electrolytes, resistance of {see 
Resistance) . 

Electromagnetic units, 11, 325. 

Electrometer, electrolytic resist- 
ance by, 115. 

Electromotive force, 13, 176; 
comparison of, by condenser 
method, 188; by galvanometer 
in shunt, 186; by the Rayleigh 
method, 189 ; by rapid charge 
and discharge, 192 ; of stand- 
ard cell by Kelvin balance, 
193 ; by silver voltameter, 
196. 

Electrostatic, units, 11; volt- 
meters, 200. 

Energy, 9 ; expended in hystere- 
sis, 299. 

Erg, 8. 

Errors of observation, effect of, 
52; in battery resistance, 101; 
in slide wire bridge, 53 ; in 
tangent galvanometer, 120. 



340 



INDEX. 



Exchanging coils, apparatus for, 
70. 

Fall of potential, resistance by^ 
95. 

Farad, 16; international, 18. 

Faraday, 157, 275. 

Fessenden, temperature coeffi- 
cient of copper, 23. 

Figure of merit of galvanometer, 
37. 

Fitch, mercnrous chloride cell, 
183. 

Fleming, electrical resistance, 
U. 

Force, 7. 

Formulas, dimensional, 1. 

Foster, Carey, method of com- 
paring resistances, 64; meas- 
uring mutual induction, 268. 

Fundamental and derived 
units, 1. 



Galvanometer, ballistic, 207 ; 
constant of ballistic, 88, 309, 
310, 318; cahbration of, 37, 
88, 150, 151, 154, 309, 310, 318 ; 
cosine, 126; d'Arsonval, 135; 
deflections corrected, 37, 211, 
321, 322; differential, 40, 44: 
figure of merit of, 37 ; in 
shunt, comparison of E.M.F.'s 
by, 186; mirror, reflecting, 31, 
34; resistance by means of 
tangent, 29 ; resistance by 
Thomson's method, 56 ; tan- 
gent, 29, 118; Thomson, 145. 

Gauss, 8. 

German silver, temperature co- 
efficient of, 23. 



Glazebrook, and Skinner, E.M.F. 
of standard cell, 196 ; appara- 
tus for exchanging coils, 72. 

Gott's method of comparing 
capacities, 219. 

Gray, determination of &6, 291. 

Guilleaume, electrical standards, 
17. 

H-form of standard cell, 184. 

Heaviside's method with the 
differential galvanometer, 44. 

Helmholtz, von, calomel cell, 
183; electrical standards, 17. 

Henry, the, 18. 

High resistance, discharge of 
a condenser through, 223 ; 
method of comparing E.M.F.'s, 
186 ; of measuring battery re- 
sistance, 98. 

Himstedt, ratio of units, 12. 

Hopkinson's bar and yoke 
method, 314. 

Horizontal intensity of the 
earth's field, 287,-288. 

Horse-power, 9. 

Houston, residual magnetiza- 
tion, 281. 

Hysteresis, magnetic, 299, 311. 

Impedance, 237; method of 
measuring self-induction, 243. 

Induced magnetization, correc- 
tion for, 295. 

Induction, magnetic-, 276, 279, 
295; mutual (see JSIutual in- 
ductance) ; self- {see Self-in- 
ductance) ; unit of, 18. 

Infinity plug, 50. 

Insulation resistance (see Ke- 
sistauce). 



INDEX. 



341 



Intensity of magnetization, 11, 
277, 295. 

Internal resistance of batteries 
{see Resistance) . 

International, ampere, 17; cou- 
lomb, 18; farad, 18; ohm, 17; 
units, 17, 18 ; volt, 18. 

Ions, 156. 

Jager, Weston standard cell, 184. 
Joule, the, 8, 18. 

Kahle,E.M.r. of Clark cell, 180. 

Kelvin, Lord {see also Thom- 
son), 214; balances, 141; 
multicellular voltmeter, 203 ; 
siphon recorder, 136. 

Kennelly, residual magnetiza- 
tion, 281 ; temperature coef- 
ficient of copper, 23. 

Kerr, optical magnetic phenom- 
ena, 299. 

Known potential differences, in- 
sulation resistance by, 83. 

Known resistances, calibration 
of galvanometer by, 154. 

Kohlrausch, conductivity of 
electrolytes, 110; determina- 
tion of BS, 291 ; magnetic 
dip, 284; resistance of elec- 
trolytes, 113; vessels. 111. 

Kupffer, density of water, 6. 

Lag, angle of, 236, 238. 
Lamp and scale, 34, 148. 
Laplace, metre des archives, 5. 
Laws of resistance, 20. 
Leakage, insulation resistance 
by, 87, 92; magnetic, 315, 319. 
Least error, 52, 53, 101, 120. 
Legal ohm, 19. 



Length, unit of, 4. 

Lindeck, temperature coefficient 

of German silver, 23 ; of nian- 

ganin, 24. 
Logarithmic decrement, 211. 

Magnet, control, 32, 148. 

Magnetic, and electrical units, 
9; axis, 275; dip, 282, 284 
field, 10, 13; on axis of coil 
122, 278; strength of, 276 
within long solenoid, 278 
flux, 276; hysteresis, 299, 311 
inclination, 282, 284; indue 
tion, 276, 279, 295; leakage 
315, 319; moment, 10, 277 
permeability, 280, 295 ; poles 
9, 275; reluctance, 315, 320 
shell, 12; susceptibility, 280, 
295. 

Magnetism, 275. 

Magnetization, correction for 
induced, 295; curves, 296, 298, 
305, 311, 316, 319; intensity 
of, 11, 277, 295. 

Magnetometer, 300. 

Magnetometric method, 299. 

Manganin, temperature coefli- 
cient of, 24. 

Mass, unit of, 6. 

Maxwell, on dimensional formu- 
las, 2 ; electromagnetic theory 
of light, 11; magnetic dip, 
284. 

Maxwell's method of comparing 
capacity and self-induction, 
245; mutual inductances, 265; 
mutual and self-inductances, 
272 ; self-inductances, 255 ; 
rule for bridge connections, 
48. 



342 



INDEX. 



Mecbain, metre des archives, 5. 

Meikle, copper voltameter, 162, 
163. 

Metre and foot, relation of, 5. 

Michelson, velocity of light, 
12. 

Miller, density of water, 6. 

Mirror, concave, in galvanome- 
ter, 35; galvanometers, 31, 34, 
145. 

Mixtures, comparison of capaci- 
ties by method of, 222. 

Momentum, 8. 

Multiplying power of shunt, 32. 

Mutual inductance, 235, 261; 
Carey Foster method of meas- 
uring, 268; comparison of , 265, 
266; comparison with self -in- 
ductance, 272. 

Newcomb, velocity of light, 12, 
Niven's method of comparing 
self -inductances, 258. 

Oersted's electromagnetic dis- 
covery, 12. 

Ohm, the, 16; international, 17; 
"legal," 19. 

Ohm's law, 15 ; calibration of 
gah^anometer by, 151. 

One pole magnetometric method, 
301. 

Optical method of magnetic 
measurement, 299. 

Paris Congress of 1881, practical 
units of, 16. 

Pendulum apparatus for con- 
denser methods, 106. 

Permeability, magnetic, 280, 295. 

Permeameter, Thompson's, 307. 



Perry's, Ayrton and, method of 
measuring electrolytic resist- 
ance, 115. 

Platinoid, 24. 

Plotting, currents, 121; BSoB 
curves, 296, 298, 305, 311, 316. 

Pohl's commutator, 28. 

Pole, strength of, 9, 276. 

Post-Office resistance box, 48. 

Potential differences, 14; meas- 
urement of resistance by, 39, 
83. 

Practical electrical units of the 
Paris Congress, 16; of the 
Chicago Congress, 16, 328. 

Preparation of materials for 
Clark cells, 176, 330. 

Quantity, 13, 207. 
Quartz fibres for galvanometers, 
36. 

m 

j Rapid charge and discharge, 

'■ comparison of E.M.F.'s by, 192. 

Rayleigh method of comparing 

E.M.F.'s, 189. 
Reduction factor by electrolysis, 
164, 329. 
I Reflecting galvanometer, 31, 34, 
\ 145. 

' Reichsanstalt, Weston standard 
cell, 184 ; standards of resist- 
ance, 174. 
Reluctance, magnetic, 315, 320. 
Residual discharges, 225 ; mag- 
netization, 280. 
Resistance, 14, 20 ; of batteries, 
96, 98, 100, 104, 106, 118; box, 
25; Post-Offlce, 48; Carey 
Foster method, 64; by differ- 
ential galvanometer, 40, 44 ; of 



INDEX. 



343 



electroh'tes, 109, 113, 115; by 
fall of potential, 95 ; of a gal- 
vanometer, 56; insulation, 83, 
SQ, 87, 92; laws of, 20; by 
potential diflerences, 39; by 
Post-Office box, -18 ; specific, 
22; standard, QQ, 68, 72, 174; 
by tangent galvanometer, 29 ; 
temperature coefficient of, 23, 
80. 

Reversals, demagnetization by, 
302; method of, 297, 311. 

Rimington's modification of 
Maxwell's method, 253. 

Ring, divided, 305 ; magnetic 
tests of, 305, 308. 

Rod, divided, 306; magnetic 
tests of, 299, 306, 307. 

Rosa, ratio of units, 12. 

Rowland, method of magnetic 
measurements, 308 ; ratio of 
units, 12. 

Russell's modification of Max- 
well's method, 251. 

Sahulka, capacity of electro- 
static" voltmeter, 240. 

Searle, ratio of units, 12. 

Self-inductance, 235 ; a length, 
248 ; comparison of capacity 
with, 245, 249, 251, 253; of 
mutual inductance with, 272 ; 
of two self -inductances, 255, 
258 ; impedance method of 
measuring, 243 ; standard of, 
257; three voltmeter method 
of measuring, 244. 

Shunt box, 34 ; multiplying | 
power of, 32. 

Siemens electrodynamometer 
127. 



Silver voltameter, 158, 196, 328; 
E.M.F. of standard cell by, 196. 

Sine inductor, 114. 

Siphon recorder, 136. 

Skinner, Glazebrook and, E.M.F. 
by silver voltameter, 196. 

Slide wire bridge, 51, 54, 56, 58, 
64. 

Solenoid, compensating, 301 ; 
field within, 278. 

Solenoidal magnetization, 277. 

Specific resistance, 22. 

Standard cell, Carhart- Clark, 
181,324; Clark, 18, 176, 324, 
330; calibration of voltmeter 
by, 205 ; combination for zero 
coefficient, 184 ; current meas- 
ured by, 169, 172; E.M.F. by 
Kelvin balance, 193; E.M.F. 
of by silver voltameter, 196 ; 
one volt calomel, 183 ; tem- 
perature coefficient of, 180, 
182 ; Weston, 184. 

Standard, condensers, 213; of 
self-induction, 257 ; resist- 
ances, Q>Q, 68, 72, 174. 

Static capacity of coils, 115, 244. 

Strength, of current, 12 ; of field, 
10, 122, 276, 278; of pole, 9, 
276. 

Sunlight, efiect on hard rubber, 
28. 

Susceptibility, magnetic, 280, 
295. 

Tangent galvanometer, 118. 

Telescope and scale, 34. 

Temperature coefficient, of re- 
sistance, 23, 80; of E.M.F. of 
standard cells, 180, 182, 183, 
185, 324, 330. 



344 



INDEX. 



Thompson's permeameter, 307. 

Thomson, J. J., ratio of units, 
12. 

Thomson (Sir Wm.), galvanom- 
eter, 115 ; ratio of units, 12 ; 
siphon recorder, 136. 

Thomson's method of galva- 
nometer resistance, o^ ; of 
mixtures, 222. 

Three voltmeter method of 
measuring self-induction, 244. 

Time constant, 248 ; is a time, 
248. 

Tractional method, 303, 305, 306, 
307; compared with ballistic, 
305, 316. 

Trallis, density of water, 6. 

Tuning-fork method, of com- 
paring E.M.F.'s, 192 ; of meas- 
uring capacity, 229, 230. 

Unit, magnetic held, 10 ; pole, 
10, 276. 

Units, dimensions of, 7, 325; 
electromagnetic and electro- 
static, 11; fundamental and 



derived, 1 ; magnetic and elec- 
trical, 9. 

Velocity, 7 ; of light, 12. 
Vertical component of earth's 

field, 297. 
Volt, 16; international, 18, 330. 
Voltameter, copper, 161 ; silver, 

158, 196, 328. 
Voltmeter, and ammeter method 

of measuring resistance, 95, 

96 ; calibration of, by standard 

cells, 205 ; electrostatic, 200 ; 

capacity of, 240 ; multicellular, 

203 ; Weston, 203. 

Wachsmuth, Weston standard 
cell, 184. 

Watt, 9, 18. 

Wattmeter, 132. 

Weber's earth-baductor, 284, 309. 

Weston instruments, 134, 136, 
203 ; standard cell, 184. 

Wheatstone's bridge, 45 ; Max- 
well's rule for, 48. 

Yoke, Hopkinson's bar and, 314. 



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